jmc

Our triangle is as shown below.



Since $\tan M=\frac{5}{4}$, we have $\frac{NO}{OM}=\frac{5}{4}$, so $$NO=\frac{5}{4}OM=\frac{5}{4}\cdot 8=10.$$Then, from the Pythagorean Theorem, we have $$\begin{array}{rl}MN& =\sqrt{N{O}^2+O{M}^2}\\ & =\sqrt{10^2+8^2}=\sqrt{164}=\boxed{2\sqrt{41}}.\end{array}$$