Probability of angular momentum

You have to express $hat L_x$ and $hat L_y$ in terms of operators whose action on $|psirangle$ you know about. Ideally, you'd write $hat L_x$ and $hat L_y$ in terms on $hat L_z$ and $hat L^2$, but usually it's easier to use the ladder operators (as suggested in the comments):

$$hat L_+ = hat L_x+mathrm{i}hat L_y, $$ $$hat L_- = hat L_x-mathrm{i}hat L_y, $$

so that

$$hat L_x = frac{1}{2} left ( hat L_+ + hat L_- right ), $$ $$hat L_y = frac{1}{2mathrm{i}} left ( hat L_+ - hat L_- right ). $$

The ladder operators' action on eigenstates of $hat L_z$ is summarised here.