Is the wife sinful after being subjected to humiliation and beatings by her husband, following her attempts to apologize to him and his family, and his refusal to pray?

The husband should accept his wife's apology. There is no sin upon her if she apologizes and he does not accept her apology, or if she abandons him after he unjustly abandoned her.

If the husband is negligent in performing prayers, the wife should strive to help him repent and become righteous by reminding him of Allah and encouraging him to associate with righteous people, so they can help him remember Allah and maintain his obligatory duties, especially the five daily prayers in congregation, for indeed, "prayer prohibits immorality and wrongdoing."

If the husband repents and maintains his prayers, the wife should live with him honorably. However, if the wife has exerted her utmost effort and her husband does not become righteous, she has the right to separate from him, but she should weigh between remaining and separating from him and choose the lesser of the two harms.