The distance from the center $(-6, -7)$ to the inside point $(-6, 5)$:

$d_1 = \sqrt{(-6-(-6))^2 + (5-(-7))^2} = \sqrt{0 + 144} = 12$

Since $(-6, 5)$ is inside the circle: $m > 12$.

The distance from the center $(-6, -7)$ to the outside point $(8, -7)$:

$d_2 = \sqrt{(8-(-6))^2 + (-7-(-7))^2} = \sqrt{196 + 0} = 14$

Since $(8, -7)$ is outside the circle: $m < 14$.

So $12 < m < 14$, and $m$ is an integer, therefore $m = \mathbf{13}$.