Is the divorce that took place Islamically valid, and what is the ‘iddah (waiting period) resulting from it, given that the wife is the one who requested the divorce and signed for it?

If the husband utters the divorce or writes it with the intention of divorce, the divorce occurs. If it is without compensation, it is a revocable divorce (ṭalāq rajʿī), and the husband can take his wife back during her waiting period (iddah). As for divorce with compensation, it is a khulʿ (dissolution of marriage initiated by the wife), and it entails a minor irrevocable divorce (baynūnah ṣughrā), after which the woman cannot return to her husband except with a new contract. The waiting period (iddah) for a divorced woman who menstruates is three menstrual cycles, and for one who does not menstruate, it is three months. If the waiting period expires and both parties desire to return to each other, it is permissible with a new contract that fulfills the conditions, including the declaration of the marriage and not keeping it secret.