For hydrogen like atom, find velocity from potential energy
The fault in your solution is the part where you assume  $\mathrm{\frac{-PE}{2} =KE}$ this is true only if the force follows inverse square law.

In this particular case we need to find the force first by using

$$F =\frac{-dU}{dr}$$

Which gives us

$$F =\frac{-3Ke^2}{r^4}$$

Now this force is central and provides centripetal force

$$\frac{mv^2}{r} =\frac{3Ke^2}{r^4}$$ 

This along with 
$$mvr=\frac{nh}{2\pi}$$

gives us
$$v=\frac{n^3h^3}{24Ke^2\pi^3m^2}$$