jmc

Set one side of the patio equal to $a$ and the other equal to $b$, producing two equations: $$\begin{array}{rl}ab& =180,\text{ and}\\ 2a+2b& =54.\end{array}$$The second equation can be rewritten as $b=27-a$. Substituting, we have $$\begin{array}{rl}180& =a\left(27-a\right)\Rightarrow \\ 180& =27a-a^2\Rightarrow \\ -180& =a^2-27a\Rightarrow \\ 0& =a^2-27a+180\Rightarrow \\ 0& =\left(a-12\right)\left(a-15\right).\end{array}$$So $12$ feet and $15$ feet are the lengths of the two sides of the patio. Therefore, the diagonal is $\sqrt{12^2+15^2}$, or $\sqrt{369}$. Therefore, the length of the diagonal squared is $\boxed{369}$.