jmc

First we find Jill's speed in miles per hour by dividing total distance by time, in which we can cancel a common factor: $$\begin{array}{rl}\text{Jill’s speed}& =\frac{x^2-3x-54}{x+6}\Rightarrow \\ & =\frac{(x-9)(x+6)}{x+6}\Rightarrow \\ & =(x-9).\end{array}$$Now we set the two speeds equal to each other and solve for $x$: $$\begin{array}{rl}x-9& =x^2-11x-22\Rightarrow \\ 0& =x^2-12x-13\Rightarrow \\ 0& =(x+1)(x-13).\end{array}$$If $x=-1$, we'd get a speed of $-1-9=-10$ miles per hour, which isn't possible. That means $x=13$, so their speed was $13-9=\boxed{4}$ miles per hour.