Honestly, if you feel you have already spoken to her about her performance and are NOT making any progress, then it is time to move on. While I can tell you feel somehow guilty giving up on her, you need to keep mindful that ultimately she is your employee. You are not happy with the work she does, despite talking with her. It also appears the previous family who dumped her on you had similar problems.

Most importantly, there is a safety issue with your kids? This would be grounds for termination, regardless of everything else in my book.