Is the wife sinful if she remains separated from her mentally ill husband, who is cared for by his brother, while she cares for her children, and is not considered a real wife due to the health, legal, and financial circumstances?

It is permissible for spouses to remain separated without divorce, but it should not be done if there is no legitimate religious justification for it, because the principle is either to hold together with kindness or to release with goodness. If the husband is insane and the wife wishes to be patient with him, she may do so and there is no sin upon her. However, if she fears harm, she may refer the matter to the judge and request a divorce. The husband's mental illness and its degree should be ascertained, and the children should not be prevented from seeing him unless it is for their benefit. As for divorcing an insane person, there are details concerning that.