Here is a strategy outline, it seems like no fun to execute due to the complicated function. We know (because $f$ is analytic and written funny) that $u_x = -v_y$ and $u_y = v_x$. We can compute $u_x - v_x$ and $u_y - v_y$ using the explicit formula. So we have four linear equations in the four "variables" $u_x, u_y, v_x, v_y$ (whose coefficients are functions), so you can solve for them.
Now if you know $u_x$ and $u_y$ you can recover $u$ up to a constant (integrate $u_x$ with respect to $x$, you'll get something well-defined up to a function of $y$, now differentiate with respect to $y$ to figure out that function of $y$ up to a constant); same for $v$. To solve for those two unknown constants, you can use your explicit value of $f(\pi/2)$ much as you would when integrating in a first calculus course.