I can't see exactly what the problem is. Is it possible the SIGCHLD is not breaking you out of the sleep?
It must boil down to one of three things:
1. Your child process isn't dying
2. The child process dies, but the parent isn't reaping it.
3. The child process dies, the parent reaps it, but %child_pids is not clear

I suggest you find out which, using ps, or similar. Most likely it's 2 or 3. Put print statements into the reaping sub; also, change the parent's sleep to sleep 1, and print the %child_pids hash every time round the while loop.