Nose brakes can help the aircraft in braking and it can also help in reducing the travel on the runway length. So why don't we use brakes in the nose landing gear?
The assumption is wrong. Some planes indeed have nose wheel brakes.
The reason why so few planes have nose wheel brakes is that
The braking force of a brake cannot exceed the weight carried by the corresponding wheel multiplied with the friction coefficient. A nose wheel carries much less weight than a main wheel, because the main wheels are much closer to the center of gravity. There is more force on the nose wheel during deceleration, but still much less than the main wheels carry.
The nose wheel brake itself, as well as stronger structure, mechanisms and wheels would add more weight to the nose of the aircraft.
Nose wheel braking reduces the stability of the decelerating plane. This could be controlled in various ways at the cost of increased complexity.
For the aircraft designer it is a trade off: is is worth extra cost, weight and complexity for reducing the braking distance by a small amount?
Note that there are planes where even some of the main gears are not equipped with brakes, for instance the A380, where no more brakes are required to fulfill the requirements.
A practical answer for a single pilot plane looks to me like a lack of ways to control a 3rd brake. During a landing the pilot's already using these controls:
- Left foot on left rudder & brake
- Right foot on right rudder & brake
- One hand on the stick
- One hand on the throttle or flaps
Unless there's going to be some type of squeeze lever on the stick (like a bicycle brake) you've run out of appendages to operate another brake.
And slaving a nose wheel brake to one of the other main brakes could be dangerous if the nose wheel starts to skid and you lose directional control.