What is the Islamic ruling on dealing with my niece, my brother's daughter, who married a Christian who did not become Muslim, and has had offspring by him, given that her parents approve of this? And how should I handle maintaining kinship ties with her when I do not accept her marriage?

The marriage of your niece to a Christian is null and void, and considered fornication. They must be separated, and any child born from this union is not attributed to him but to its mother, as it is the result of an illicit act. Maintaining kinship ties with her is obligatory for the purpose of advising her and reminding her of God, for she may repent. Abandoning her is not permissible unless it is strongly believed that it would be beneficial. We also draw attention to the seriousness and ugliness of the stance taken by the girl's parents, as they consented to their daughter's marriage to a disbeliever and her bearing children from him without disapproval. This constitutes a betrayal of trust and an abandonment of responsibility.