Does this price elasticity depend on p?
That type of demand is called an Isoelastic demand function, because the elasticity of demand is constant, so it does not depend on prices.

You have not done any mistake. But you just need to move forward one more step. Notice that at the moment your derivative has both p and q in it. Try to get rid of q. This is, try to have $\epsilon=f(p)$, and not $\epsilon=f(p,q)$. This hint should be enough for you to solve the question.