The aviation enthusiasts to whom this letter was address in 1905 would have understood the Wright brothers to be referring primarily to special light-weigh steam engines specifically designed for use in aircraft. Enthusiasts had been building such engines since at least the 1840’s and by the 1870's you could have one built for you if you could afford it.
The statement made by the Wright brothers has two parts: 1) It is a mistake to conclude that 19th-century aero-engines had insufficient power for flight simply because the flying machines in which they were mounted failed to fly, and 2) suitable engines existed 50 years ago. The first statement is almost certainly true. The second statement is true only with the major qualification that the engines actually build before the 1870's seem to have been small-scale prototypes.
In 1905 when the Wright brothers wrote this letter the only working powered airplanes (other than their own) are models too small to carry a pilot. As far as the Wright brothers know nobody has been able to get a man-carrying airplane more than a meter into the air, keep it there for more than a few seconds, and travel more than 100 meters along the ground.
We now now that there are at least six principal reasons why 19th century airplanes did not work: 1) The wings had a poor lift to drag ratio, 2) The propeller blades did not have the shape they would need to provide good thrust for the input power, 3) the airplanes lacked three-axis control, 4) the controls they did have were difficult to operate, 5) the operators did not know how to fly an airplane, and 6) the steam engines were not powerful enough to overcome problems one and two.
The Wright brothers are reporting that they now have an airplane which actually flies. They have achieved this by attacking problems one through five. If you solve all of them, you do not need to solve problem six.
Their core claim about airplane engines is expressed in numbers. They say that with their design improvements an airplane can fly with an engine which produces only one horsepower for every 30 kg of gross vehicle weight. This is an approximation of what they actually achieved on December 17, 1903 at Kitty Hawk with an airplane which weighted 338 kg and a 12 hp engine (28 kg/hp).
This claim is also in line with what other researchers had already discovered. The Wright brothers had read Octave Chanute’s 1894 book Progress in Flying Machines in which he reports (p. 73) tests of airplane wings (not entire airplanes). Professor Langley found that a wing could lift 200 lbs (90 kg) per horsepower while Mr. Maxim’s tests yielded a figure of 133 lbs (60kg).

The 1903 Wright engine as seen in 1916 after restoration
The homemade 12 hp engine used by the Wright brothers weighed 170 lbs dry. With fluids it weighted about 200 lbs. In 1903 it produced between 12 hp continuously and 16 hp peak (7.5 and 4.6 kg/hp respectively). This is very modest performance for a gasoline engine. For comparison the Clerget 9B used in the Sopwith Camel in 1917 weighted 173 kg (dry) and produced 130 hp which is 1.33 kg/hp (slightly worse with fluids of course).
While it is difficult to find reliable and detailed information about the performance of 19th century steam aero-engines, it is clear that a number of them had power to mass ratios equal to or better than that of the Wright engine.

1843: William Hensen and John Stringfellow obtained a British patent for the concept of an "Aerial Steam Carriage". This was to be a passenger airliner with a 150 foot wingspan driven by a 50 hp light-weight steam engine. They sought investors and began experiments with a model with a 20-foot wingspan and a working light-weight steam engine. They conducted tests between 1843 and 1847. Though the engine worked, making the even a large-scale model strong and stiff enough to fly outdoors proved beyond their ability.
Some notes from a bench test of the engine have been preserved:
June, 27th, 1845, water 50 ozs., spirit 10 ozs., lamp lit 8.45, gauge moves 8.46, engine started 8.48 (100 lb. pressure), engine stopped 8.57, worked 9 minutes, 2,288 revolutions, average 254 per minute. No priming, 40 ozs. water consumed, propulsion (thrust of propellers), 5 lbs. 4 1/2 ozs. at commencement, steady 4 lbs. 1/2 oz., 57 revolutions to 1 oz. of water, steam cut off one-third from beginning. The diameter of cylinder of engine was one-and-a-half inch, length of stroke three inches.
Note that it took three minutes to get up steam and the engine ran for nine minutes. If we estimate the average pressure in the cylinders at 50% of the boiler pressure, this engine produced about 1/6 hp.

Photograph of Stringfellow's 1848 model airplane

Photograph of one of the numerous surviving Stringfellow aero-engines
1848: John Stringfellow builds a smaller model with a 10-foot wingspan and a new engine. According to Chanute, Stringfellow tested it indoors in a room less than 66 yards long. It ran one third of the length of the room along a guide wire. After leaving the wire it would continue to fly to the end of the room. If would frequently rise after leaving the wire at a rate as high as one in seven. The weight of the entire model including engine, water, and fuel was less than 6 ½ lbs.
In 1995 a reproduction of Stringfellow's 1848 model was build for the BBC2 program Local Heroes. This model was tested in the same mill Stringfellow had used. A published advertisement for the program implies that the model flew. I would like to know more about this program.

According to the Smithsonian this is the Stringfellow engine which won a Royal Aeronautical Society prize for best power to mass ratio in 1868. It is unclear when the engine was built. Stringfellow wrote he had removed
it from an unspecified "old model" to take it to the exhibition.
1868: The Royal Aeronautical Society held a show in the Crystal Palace in London. Due to his reputation, John Stringfellow was invited to exhibit one of his models. He got them out of storage but decided none was in sufficiently good condition and so built a new tri-plane model with a 1/3 hp steam engines. During the exhibition it ran along a wire stretched across the exhibition hall. Free flight was not allowed due to the fire risk, but the machine seemed to lift the wire.
Stringfellow also exhibited a steam engine removed from one of his old models. The cylinder was two inches in diameter, the stroke length three inches. The working pressure 100 psi. It drove two four-blade propellers, three feet in diameter, at 300 rpm. As reported in Is a Flying Machine a Mechanical Possibility? (Scientific American, March 13, 1869, volume 20, Number 11, p. 169) the power was estimated as follows:
Area of piston, 3 inches; pressure in cylinder, 80 lbs. per square inch; length of stroke, 3 inches; velocity of piston, 150 feet per minute; 8 x 80 x 150 = 36,000 foot-pounds. This makes rather more than one-horse power (which is reckoned at 33,000 foot-pounds). The weight of the engine and boiler was only 13 lbs., and it is probably the lightest stream engine that has ever been constructed.
(This calculation seems to imply that the cylinder was double-acting. It also estimates that the average pressure in the cylinder was 80% of the boiler pressure which seems generous.)
If this engine used twice as much water and fuel as the 1845 engine (which was filled with 50 oz of water and 10 oz of fuel), that would add about 1.6 kg to the weight for a total of about 7.5 kg. That works out to 6.8 kg/hp, about the same as the Wright engine, albeit at a smaller size. Mr. Stringfellow was awarded a prize for the best power to weight ratio yet achieved in a steam engine.
1874: Félix and Louis du Temple tested a monoplane powered by a steam engine of their own design. The engine was equipped with a boiler in which small pipes were tightly packed into a lightweight firebox. According to Chanute (Progress in Flying Machines, 1894, pp 90-92) the machine, which was intended to carry a man, had a wingspan of 40 feet and weighted about 160 lbs. According to Sobolyev (История самолетов мира, 2001, p. 27) the engine weighted 59 kg. If it developed the 6 hp some sources suggest, then that is 9.83 kg/hp, worse than the engine in the Wright Flyer.

Plans for Mozhaychy's 1884 Monoplane

Drawing of the 10 hp engine which Messrs. Ahrbecker, Son & Hamkens, of Stamford Street, S.E. made for Captain Mozhaysky
In 1884 Russian naval officer Alexander Mozhaycky tested a steam-powered monoplane with two stream engines imported from England. According to Sobolyev (История самолетов мира, 2001, p 29) one engine produced 10 hp, the other 20 hp. He gives the combined mass to power ratio of the entire propulsion system including both engines, boiler, condenser, and separator as 5.5 kg/hp which is better than the performance of the Wright engine as measured in 1903. Sobolyev (p 30) quotes a report (Военная энциклопедия. Т16, Спб., 1916. С. 377) which reads “The apparatus rose from the ground, but being unstable it tipped onto its side and broke a wing”.
The engines were described in the Scientific American Supplement for the years 1881 and 1882 as follows:
The larger of these engines has cylinders 3¾ in. and 7½ in. in diameter and 5 in. stroke, and when making 300 revolutions per minute it develops 20 actual horse power, while its weight is but 105 lbs. The smaller engine--the one illustrated--has cylinders 2½ in. and 5 in. in diameter, and 3½ in. stroke, and weighs 63 lbs., while when making 450 revolutions it develops 10 actual horse power.
The two engines are identical in design, and are constructed of forged steel with the exception of the bearings, connecting-rods, crossheads, slide valves and pumps, which are of phosphor-bronze. The cylinders, with the steam passages, etc., are shaped out of the solid. The standards, as will be seen, are of very light T steel, the crankshafts and pins are hollow, as are also the crosshead bolts and piston rods. The small engine drives a single-acting air pump of the ordinary type by a crank, not shown in the drawing. The condenser is formed of a series of hollow gratings.
Steam is supplied to the two engines by one boiler of the Herreshoff steam generator type, with certain modifications, introduced by the designers, to insure the utmost certainty in working. It is of steel, the outside dimensions being 22 in. in diameter, 25 in. high, and weighs 142 lb. The fuel used is petroleum, and the working pressure 190 lb. per square inch.
1890: Clement Ader tested a steam-powered monoplane with bat-like wings. It was equipped with a 20 hp engine weighing 51 kg (2.55 kg/hp). With a gross weight of 330 kg the machine weighed the same as the 1903 Wright Flyer, but had 67% more engine power. Ader’s machine rose about 25 cm off the ground and flew for about 50 meters.

Sir Henry Maxim's test rig which was damaged on July 31, 1894 when the captive flying machine lifted and tore itself off of the track before the dual 180 hp steam engines could be stopped

Sir Henry Maxim holds one of his 180 hp steam aero-engines

1894: Sir Henry Maxim operated a test rig for experiments with various wing configurations. In the April 1910 issue of the Journal of the Western Society of Engineers Octave Chanute described the experiments as follows:
Mr. Maxim built an enormous apparatus, weighting 8,000 pounds and spreading 4,000 feet of surface, moved by a steam engine of 360 horsepower. That machine was run upon a track of 9 feet gauge a good many times, and on one occasion it undertook a vagabond flight on its own account; its equilibrium was bad, however, and the steam was shut off; the machine alighted somewhat broken.
According to Sobolyev (История самолетов мира, 2001, p. 34) the engines, boiler, and condensers together weighted about a tonne for a mass to power ration of 3.1 kg/hp. At 8,000 lbs gross weight and 360 horsepower the Maxim test rig had one horsepower for every 10 kg of gross weight. That is nearly three times as much engine power per unit of vehicle mass as the 1903 Wright Flyer had.
Maxim, in his thesis Natural and Artificial Flight (Excerpted in The Aeronautical Annual, 1896, published as Artificial and Natural Flight in 1908) describes his experiments in detail and concludes:
My experiments have certainly demonstrated that a steam engine and boiler may be made which will generate a horsepower for every six pounds of weight, and that the whole motor, including the gas generator, the water supply, the condenser, and the pumps may be all made to come inside of 11 lbs. to the horse-power. They also show that well made screw propellers working in the air are fairly efficient, and that they obtain a sufficient grip upon the air to drive the machine forward at a high velocity; that very large aeroplanes, if well made and placed at a proper angle, will lift as much as 2 and a half lbs. per square foot at a velocity not greater than 40 miles an hour; also that it is possible for a machine to be made so light and at the same time so powerful that it will lift not only its own weight but a considerable amount besides, with no other energy except that derived from its own engines. Therefore there can be no question but what a flying machine is now possible without the aid of a balloon in any form.
So after experiments this respected aviation pioneer concluded that it was perfectly possible to build aero-engines weighing less than 5 kg/hp and that a steam-powered airplane was definitely possible.
1897: Samuel Pierpont Langley, secretary of the Smithsonian, achieved sustained powered flight with two large models equipped with steam engines which ran for up to two minutes. Interestingly in 1889 Langley had bought at least one Stringfellow engine from the inventor's son and sent it off for study.
In conclusion, when the Wright brothers wrote this letter they could allude to a well-known history of steam aviation engines stretching back more than 50 years. The steam aero-engines actually installed in actual man-carrying airplanes build and tested in the last quarter of the 19th century likely had a power to mass ratio comparable to that of the homemade gasoline engine used by the Wright brothers in their 1903 Flyer. While one can find fault with the "50 years ago" part of the statement, the overall statement that 19th century aero-engines were sufficient for sustained flight is sound.