What is the waiting period (iddah) required of a wife after khulʿ, knowing that she had been separated from her husband for over a year before he signed the khulʿ papers?

First, the waiting period (iddah) begins from the time of divorce or khul‘ (divorce at the wife's instance), even if it was preceded by a long period of estrangement, because rulings are linked to their causes.

Second, if the spouses agree that the khul‘ should become effective after two months, then the khul‘ occurs at the end of the two months, and the waiting period begins from that time.

Third, scholars differed concerning the waiting period for khul‘. The majority of them hold that it is the same as the waiting period for divorce (three menstrual cycles or three months). Others held that it is one menstrual cycle, which is the view of some of the Salaf and was chosen by Ibn Taymiyyah and Ibn al-Qayyim. You must observe what the Sharia judiciary in your country mandates regarding these disputed matters.