Solution:

Since $5^2 + 12^2 = 13^2,$ the triangle is right. Place it at $(0, 0),$ $(5, 0),$ $(0, 12).$ The circumcenter is the midpoint of the hypotenuse, $\left(\tfrac52, 6\right).$

The inradius satisfies $(12 - r) + (5 - r) = 13,$ so $r = 2$ and the incenter is $(2, 2).$

The distance is $\sqrt{\left(\tfrac52 - 2\right)^2 + (6 - 2)^2} = \sqrt{\tfrac14 + 16} = \dfrac{\sqrt{65}}{2}.$

Thus, the correct answer is D.