Praise be to Allah.
Firstly: we ask Allaah to help your sister to deal with this calamity and to bring her relief, and to guide her husband to give up this major sin in which he is persisting.
With regard to the ruling on her staying with him, it is permissible for her to do so if she is certain that she and her children will be safe.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: I think that if she has advised him and he has not responded, then she has the right to ask for an annulment of the marriage. But there may be reasons why she cannot annul the marriage, because she has children and annulment may cause further problems. If his sin does not reach the level of kufr, then there is no sin on her if she stays with him for fear of evil consequences. But if his sin reaches the level of kufr, such as if he does not pray, then she should not stay with him a second longer.
Al-Liqa’ al-Maftooh, Q. 518.
Secondly: Our advice to her is that she should weigh up what is in her best interests, and consult her family and relatives, who know her situation best. Then she should ask Allaah’s guidance (by praying istikhaarah), for Allaah will never let her down. As the report says: No one loses by praying istikhaarah and no one regrets consulting others. Whether she chooses to annul the marriage or to stay, let her accept what Allaah has decreed for her and be patient and seek reward. Allaah says (interpretation of the meaning):
“but give glad tidings to As‑ Saabiroon (the patient)”
And Allaah knows best.