How can one deal with a husband who taunts his wife with her bad past and the details of her repentance, while he himself is not committed to righteousness and persists in his disobedience? What is the way to solve the wife's problem given the presence of children and the fear for them?

A husband is not permitted to remind his wife of sins from which she has repented, because repentance erases the sin. The wife should thank Allah for her repentance and strive to help her husband repent. If that is not successful, then she must weigh the harm of divorce against the harm of remaining with him in that state.