What is the ruling of Islamic law regarding a couple returning to each other after khulʿ, then divorcing her in a state of extreme anger and returning to her on the same night, then divorcing her again in a state of extreme anger while she is menstruating, knowing that she suffers from a mental illness and they have seven children?

Anger does not prevent the occurrence of divorce as long as the person is aware of what he is saying. Divorcing a menstruating woman or divorcing her during a period of purity in which intercourse occurred is an innovative divorce (talaq bid'i) and is considered valid by most scholars. If you divorced your wife twice while you were aware, then your divorce is effective. As for khulʿ (divorce at the wife's instance, usually with compensation), there is a difference of opinion as to whether it is a divorce or an annulment. If it is considered a divorce, then your wife has become irrevocably divorced from you (baynunah kubra), and she is not lawful for you until she marries another husband and he consummates the marriage with her. If it is considered an annulment, then you may take her back during her iddah (waiting period) if you have not divorced her more than two times. It is recommended to refer the matter to a Sharia court or to reliable scholars.