What is the Shar'i ruling regarding a wife who is not committed to prayer and refuses to wear the jilbab, and what is the impact of this on children in the West, and what are the implications of divorcing her concerning child custody?

You are responsible for your wife before Allah, and you must enjoin her to pray and observe hijab, and continue to advise her with wisdom and good counsel. If she does not respond, then you may separate from her.

As for the custody of children in case of divorce, the mother has more right to the child until the age of seven, then the child is given the choice between their parents. As for the girl, the father has more right to her after the age of seven. The reference for matters of custody is the Sharia court.

The custodian must possess competence, protection, and care. Among the conditions for custody are: Islam and trustworthiness in religion. Since your wife abandons prayer, she is not fit to have custody of the children.