As the title states, the definite integral in question is
$$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$
where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the negative real axis.
Edit: As pointed out in a comment, this isn't an integral that should be done with a keyhole contour (edited out of my original question).
My question thus is, how does one evaluate this contour integral given that the contour is not a closed loop due to the presence of a branch cut? I am studying out of Ahlfors and integrating on a Riemann surface hasn't been introduced yet.