jmc

Let $\angle FCD=\alpha$, so that $FB=\sqrt{12^2+13^2+2\cdot 12\cdot 13\sin (\alpha )}=\sqrt{433}$. By the diagonal, $DB=13\sqrt{2},D{B}^2=338$. The sum of the squares of the sides of a parallelogram is the sum of the squares of the diagonals.$$E{F}^2=2\cdot (5^2+433)-338=\boxed{578}.$$