Ryan XV-5 Vertifan used exhaust gas from engine to rotate the fan inside the wing. If I want to design an aircraft using this propulsion system how can I calculate how much power I can get from a specific engine exhaust.
We really answer this question in 2 parts.
The first is how much thrust the engine can generate at a given airspeed, either in pure thrust from exhaust gasses, or torque transferred to a mechanically rotating device, which in turn creates thrust.
this quality is known as specific impulse
Fortunately for aircraft such as the Ryan XV-5, the F35, and helicopters, converting engine torque to vertical rotor lifting force, at lower airspeeds, can be far more efficient than simply exhausting gasses vertically.
Helicopters, such as the R-22, can generate over 10 pounds of lifting force per horsepower. Propeller driven aircraft get 3-4 lbs per horsepower flying horizontally, turbofans and turbojets less but at much higher speeds$^1$.
If you are considering a design such as the XV-5, power requirements per pound weight would be in the "ball park" of the F-35B. With a larger a rotor over the top, like this one, they would be less.
$^1$ yes, jets do generate rotating mechanical energy, or "horsepower", to turn their turbines and fans.