A triangle with sides of length $5, 12,$ and $13$ has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? $\mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad$ $ \mathrm{(B) \ } \frac{7}{2} \qquad$ $ \mathrm{(C) \ } \sqrt{15} \qquad$ $ \mathrm{(D) \ } \frac{\sqrt{65}}{2} \qquad$ $ \mathrm{(E) \ } \frac{9}{2}$