Praise be to Allah.
If the husband does not pray then it is not permissible for your sister to stay with him, because the one who does not pray is a kaafir according to the sounder of the two scholarly opinions. See the answer to question no. 5208 and 6257
If he did not pray during the marriage, then the marriage is not valid but the child is to be attributed to him because she was born as the result of a marriage which both partners thought was valid.
If he stopped praying after getting married and continued not praying until her ‘iddah ended, then the marriage is annulled. But if he repents and starts to pray, she can go back to him on the basis of a new marriage contract if she agrees to.
Some of the scholars say: if he repents and starts to pray, he may go back to her even if that is after the ‘iddah has ended, so long as she has not got married to another husband.
From this you will know that according to sharee‘ah there is no need for a divorce (talaaq) from him, but because she is regarded as his wife in official papers and that has an impact on inheritance between them and preventing her from marrying someone else, she has to try to get the divorce even if she has to give him money so that he will divorce her.
With regard to the child, the mother is entitled to custody.
The husband should be advised to repent to Allah, may He be exalted, and to pray. He should be told that his wife is not permissible for him until he starts to pray. If he repents, praise be to Allah, but if he remains as he is then your sister should try to get a divorce and she can refer the matter to the court and ask for divorce because he is not spending on her and because of the harm caused.
We ask Allah to grant her a way out.
And Allah knows best.
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