According to the sounder of the two scholarly opinions—one that is upheld by the clear proofs of the Qur’an and the Sunnah—if this woman persists in abandoning the prayer and the fast, you may not keep her under your protection. She is not a Muslim, and it is unlawful for a Muslim man to remain married to a disbelieving woman who is not from the People of the Book.

As for swearing by other than Allah, this is minor shirk (associationism) from which one must repent and which one must avoid, for the Prophet ﷺ said, “Whoever swears by other than Allah has committed disbelief or associationism,” and in another ḥadīth, “Whoever feels obliged to take an oath, let him swear by Allah, or else remain silent.”

Swearing by anything besides Allah is minor associationism. A Muslim must turn from it in repentance and restrict his oaths to Allah alone, for an oath is an act of veneration, and veneration is the exclusive right of Allah.

If, however, by “family” he means the members of his household—his brothers and parents—then he must advise them and censure their behaviour.

If they refuse to heed the advice and persist in abandoning the prayer and the fast, he must separate himself from them and shun them, for Allah the Exalted says: “You will not find a people who believe in Allah and the Last Day loving those who oppose Allah and His Messenger, even if they are their fathers, their sons, their brothers, or their kindred” (al-Mujādila 58:22).