In triangle $PQR,$ let $X$ be the intersection of the angle bisector of $\angle P$ with side $\overline{QR}$, and let $Y$ be the foot of the perpendicular from $X$ to side $\overline{PR}$. If $PQ = 9,$ $QR = 12,$ and $PR = 17,$ then compute the length of $\overline{XY}$.
