Given the harmonic function u(x,y), find its harmonic conjugate v(x,y) then express the analytics function f(z) in terms of z

Question

I was given the harmonic function $u(x,y) = \frac{y}{x^2+y^2}$. Using the cauchy riemann equations i was able to deduce that $v(x,y) = \frac{x}{x^2+y^2}$. I now have the expression $f(z) = \frac{y}{x^2+y^2} + i\frac{x}{x^2+y^2}$. I was wondering if anyone could help me simplify this in terms of z. I know that $|z|^2 = x^2 + y^2$ but I don't clearly see what I can do to $y+ix$ since z is equal to $x+iy$. Could i factor out an i and have $i(x-iy)$ which would be $\frac{i\bar{z}}{|z|^2}$ which could then be $f(z) = \frac{i}{z}$? Thanks so much