How can I solve initial velocity for a projectile given the max height and distance

Compute the time to fall from $y_{max}$ to $0$ and from $y_{max}$ to $y_0$. This gives you the time of flight. From $x_f$ you can get $V_0cos theta$

Compute the vertical speed after the fall from $y_{max}$ to $y_0$. This is the initial upward speed $V_0 sin theta$

Take the $arctan$ of the ratio of initial velocities to get $theta$. Use that to get $V_0$.