Christmas Contest - Guts Round - Set 3 Problem 1

In $\triangle ABC$, $\angle C = 90^{\circ}, AC = 5, BC = 12$. The side length of the largest square that can fit in $\triangle ABC$, where one side of the square must be on the hypotenuse, can be expressed as $\frac{m}{n}$, where $\gcd(m, n)=1$. Compute $m -n$.