Will a sleeping pill hurt your cat?

I really wouldn't try it. There's really no way of telling how a drug/medicine intended for one species will work in another species

• specially when you want to use it outside its intended purpose.

You may well end up only injuring the cat, which would be really, really bad.

If it's a healthy cat you no longer can take care of - hand it over to a local animal healthcare group(ASPCA/RSPCA or similar).

If it's an old sick cat, do the responsible thing and take it to a veterinarian.