jmc

Completing the square gives $(x-2{)}^2+(y-4{)}^2=20$, so the circle has radius $\sqrt{20}=2\sqrt{5}$ and center $(2,4)$. The distance between $(2,4)$ and $(5,-2)$ is given by $\sqrt{(2-5{)}^2+(4-(-2){)}^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}$. Hence, the shortest distance is the difference of the distance between the center and the point and the radius, yielding $3\sqrt{5}-2\sqrt{5}=\sqrt{5}$. Thus, $m=\boxed{5}$.