smc

Warning: this solution is very intensive in calculation. Please do NOT try this on the test! Let's start by finding $AH$. By Heron's Formula, $s=\frac{13+14+15}{2}=21,[ABC]=\sqrt{21\ast (21-13)(21-14)(21-15)}=84$. Using the area formula $A=0.5bh$, $AH=12$. Now using the Pythagorean Theorem, $BH=5,HC=9$. Now $AM=MB=6.5$. Using Stewart's Theorem on $△ABH$, letting $HM=x$: $13\ast 6.5\ast 6.5+13x^2=12\ast 6.5\ast 12+5\ast 6.5\ast 5$ (remember that Stewart's Theorem is $man+dad=bmb+cnc$). Thus $x=6.5$ or $x=-6.5$ (reject this solution since $x$ is positive). Thus $HM=6.5$. Select $\boxed{6.5}$. P.S.: Although this is torturous, this is a good practice of Heron's formula and Stewart's theorem though.