Solution:

First, note that $T O = G O$ as $O$ lies on the perpendicular bisector of $G T$. Then if $M$ is the midpoint of $G T$, we have that $\triangle G R T \sim \triangle G M O$, so we can compute

$$T O = G O = G M \cdot \frac{G T}{G R} = \frac{13}{2} \cdot \frac{13}{5} = \frac{169}{10}.$$