Does Khul' take place if the wife asks her husband for Khul' during a fit of extreme anger due to her psychological illness, and the husband agrees to it, then they reconcile and the wife returns to her husband, knowing that the husband does not remember his intention at the time he accepted the Khul'?

The mere act of the husband taking money and saying "okay" does not constitute a valid khula', especially if it was done to calm down a sick wife. As long as the husband does not recall his intention at that moment, the marriage is considered to remain valid and khula' is not deemed to have occurred. This is because divorce does not take place merely based on doubt, and marriage is established with certainty, thus it cannot be dissolved by doubt. Ibn Qudamah and Ibn Uthaymeen both stated that doubt regarding divorce is not taken into consideration, and the fundamental principle is the continuation of the marriage.