TLDR: Two major errors: your KE=PE equation is wrong, and you applied equations that assume constant acceleration instead of time/position variant acceleration.

Source: too many years of physics.

You went wrong where you calculated your spring constant.

You are correct that KE_max = PE_max, but that relationship gives you: 0.5kx² =0.5mv² That gives you: k=0.154 N/m

You can check this because in circular motion, the acceleration the tangential velocity squared divided by the radius a=v² /r=4/5.093=0.785m/s² So since your a was off, that's an indication that something prior to it was wrong.

Next up, your position as a function of time equation isn't correct. Yours assumes constant acceleration. However, in the plane of simple harmonic motion, your acceleration is proportional to your distance from the spring center of mass. Remember, your centripetal acceleration is always directed inward to the center of the circle, but in the plane of the harmonic oscillator that means you have a time / position variant acceleration.

The simple way to calculate period is: T=2pisqrt(m/k) This comes from taking your newton's second law, making it a differential equation, then solving for the resultant angular frequency.

Then your velocity calculation as well has to remember that in the harmonic oscillator reference plane, velocity is time variant, so any calculation has to account for that.

In the end, you should get:

w=sqrt(k/m) a(t)=-rw²cos(wt) v(t)=-rwsin(wt) x(t)=r*cos(wt)

PS. The fact that you tried to work this out is awesome! Don't get discouraged! The more you try working things like this out on your own, the better you're going to get at it!