I would just WLOG $k \geq m \geq n$, and continue with mod bash, as there are only 10 cases

The quadratic residues modulo $5$ are $0,1,4$ however $m^2\equiv0\pmod{5}\iff m\equiv0\pmod{5}$. By php, two of $m^2,n^2,k^2$ have the same residue modulo $5$. Thus, one of $k^2-m^2,m^2-n^2,n^2-k^2$ must be divisible by $5$.