I have been running in circles with this for a while now.
It seems that the only solution is $(n,m,k)=(2,3,2)$ but I don't know how to prove it.
Things I have noticed: WLOG $n\geq m$ we see that $k^2$ is a multiple of $m!$. I have a feeling that this might be key; maybe there is a restriction on squares and factorials which is only possible for $n=k=2$... I tried manipulating things, the only potentially useful expression I could come up with was:
I have tried modular arithmetic on both sides to get some additional conditions but nothing useful came up. Maybe we could make use of $m!n!<(m+n)!$ to bound $k^2?$ (I doubt it though since the RHS blows up too fast).
I don't know if any of this information is useful though.