The last thing you want is for a loose piece to snag on something and pull further down into the quick, causing more pain and bleeding.
If there is loose piece hanging off the side or end, take a pair of human nail clippers and cut the hanging piece off, CLOSE TO the break, but do not cut the actual nail any shorter. If you make the hanging piece shorter, it's less likely to catch on anything. Try to do this when the kitty is sleeping. It makes things easier.
Otherwise, if there's nothing that can be safely removed, just try to make sure the broken nail stays as clean as possible, and watch it for infection or further damage. Either one of these indicates a possible trip to the vet or groomer.
Cats naturally shed the outer sheaths of their claws, and having a good scratching surface can help them remove this outer growth, and keep their claws healthy.
I hope this helps.
