A Right Angle in the Complex Plane

Let $f(z) = z^2 - 19z$. There is a complex number $z$ such that the points corresponding to $z$, $f(z)$, and $f(f(z))$ in the complex plane are the vertices of a right triangle whose right angle is at the vertex corresponding to $f(z)$. This number can be written as $z = m + \sqrt{n} + 11\mathrm{i}$, where $m$ and $n$ are positive integers. Find $m+n$.