What is the ruling of Islamic law regarding a stepmother remaining in the same house with her wicked stepson, and what is the best way to help him repent and commit to religious observance?

There is no objection to the husband's son living with his father's wife, for he is a mahram to her. However, one who commits forbidden acts should be avoided. For his guidance, one can sincerely intend and ask Allah for his guidance, advise him with kind words, give him appropriate tapes and books that suit his personality, seek the help of someone who influences him to advise him, and if he is not married, help him to get married.