What should I do about my knowledge of my divorced sister's forbidden relationship with a foreign man, given my fear for her daughter's reputation and my mother's health, and her continued stay in my house?

It is not permissible for a Muslim woman to establish a relationship with a man who is a stranger to her. If it is proven that the sister has done so, then the news must be verified, in accordance with the Almighty's saying: "O you who have believed, if there comes to you a disobedient one with information, investigate" (Quran 49:6). If the matter is confirmed, then she must be reformed without informing the mother if there is no benefit in doing so. It is forbidden to kill the sister or her betrayer. It is advised to marry her to a righteous man to safeguard her.