Suppose we have a complete graph $K_n$ on $n$ vertices. Are there any results on the ways to cover $K_n$ with $k$ copies of $K_m$, for $m<n$, such that each edge of $K_n$ is contained in exactly two of the copies of $K_m$? I've been able to find results on clique covering and the cycle double cover problem for complete graphs, but nothing on double covering by complete graphs in particular.

It may be of note that this is exactly the cycle double cover problem for $n=4$ (in which case $m$ must be $3$, and $K_3$ is a cycle), but this falls apart once $m\geq 4$. Links to resources would be much appreciated.