Find the inverse of the function
\[f(x) = \frac{2x + 5}{x - 11}.\]
Let $y = \frac{2x + 5}{x - 11}.$  Then
\[xy - 11y = 2x + 5,\]so $xy - 2x = 11y + 5.$  Then $x = \frac{11y + 5}{y - 2}.$

Hence, the inverse function is given by
\[f^{-1}(x) = \boxed{\frac{11x + 5}{x - 2}}.\]