Companies who try to back charge employees or make them fix problems on their own time do it at their peril. It is a violation of the labor laws in most states.
You really need to have a formal warning program in place and simply let the worker go when they get their three strikes. Sparky is right you need to document the whole process. Nobody likes to get fired but if you went through the whole process, signing the warnings along the way it shouldn't be shocking to you.
