What is the Islamic legal ruling regarding a wife's refusal to return to the marital home and fulfill her husband's desires, given that she and her children suffer from the husband's mental illness and he refuses to divorce her?

It is recommended to supplicate frequently for your husband, try to convince him to be consistent with treatment, and seek help from those close to him. The basic principle is that you should reside where your husband resides, and you should not leave the marital home except in the case of real harm. The husband has the right to demand his legitimate rights, and you must fulfill his desires as long as the marriage is ongoing, unless there is a legitimate excuse. You can either be patient and live with your husband in kindness, or seek separation from him honorably if you are harmed, by convincing him to divorce or khul‘ (dissolution of marriage at the wife's instance), or by taking the matter to a religious judge. You cannot separate except through a divorce from him or a judicial ruling.