Is `Iddah Necessary after Khul`?
Question

Does iddah imply when it is the woman who is seeking the divorce (khul`)?
Praise be to Allah.What is Khul`?
Khul` – in principle – can only occur at the request of the wife, and with the husband’s subsequent agreement to end the marriage. 
When does `Iddah become obligatory?
`Iddah is obligatory upon every woman who leaves her husband, or whose husband leaves her, whether the cause is divorce, annulment of the marriage or the death of the husband, except when the divorce occurs before the marriage has been consummated, in which case the woman does not have to observe `Iddah, because Allah says (interpretation of the meaning): 
“O you who believe! When you marry believing women, and then divorce them before you have sexual intercourse with them, no `Iddah [divorce prescribed period] have you to count in respect of them” [Al-Ahzab 33:49] 
Is `Iddah Necessary after Khul`?
With regard to the `Iddah following Khul`, the correct scholarly view is that it is one menstrual cycle, as is indicated by the Sunnah. 
It was narrated from Ibn `Abbas that the wife of Thabit ibn Qays ended her marriage to her husband by means of Khul` at the time of the Prophet (peace and blessings of Allah be upon him). The Prophet (peace and blessings of Allah be upon him) commanded her to observe an `Iddah of one menstrual cycle. (Narrated by At-Tirmidhi, 1185 and Abu Dawud, 2229. It was also narrated by An-Nasa’i (3497) from the Hadith of Ar-Rabi` bint `Afra’. The two Hadiths were both classed as authentic by Ibn Al-Qayyim, as we shall see below). 
Ibn Al-Qayyim (may Allah have mercy on him) said: 
In the fact that the Prophet (peace and blessings of Allah be upon him) commanded the woman who ended her marriage to her husband by Khul` to observe an `Iddah of one menstrual cycle, there is evidence for two rulings: 
The first is that she does not have to wait for three menstrual cycles, rather one menstrual cycle is sufficient. Just as this is clearly the Sunnah, it was also the view of Amir Al-Mu’minin `Uthman ibn `Affan, `Abdullah ibn `Umar ibn Al-Khattab, Ar-Rabi’ bint Mu`awwadh and her paternal uncle, who was one of the greatest Companions. We do not know of anyone who held a different opinion, as Al-Layth ibn Sa`d narrated that Nafi` the freed slave of Ibn `Umar heard Al-Rabi` bint Mu`awwadh ibn `Afra’ telling  `Abdullah ibn `Umar (may Allah be pleased with him) that she had ended her marriage to her husband by Khul` at the time of `Uthman ibn `Affan, and her paternal uncle had come to `Uthman ibn `Affan and said, The daughter of Mu`awwadh ended her marriage to her husband by Khul` today, so should she move (from the marital home)?  `Uthman said, She should move, and there is no inheritance between them, and she does not have to observe any `Iddah, but she should not remarry until one menstrual cycle has passed, lest she be pregnant. `Abdullah ibn `Umar said: and `Uthman was the best of us and the most knowledgeable. 
This view was also shared by Is-haq ibn Rahawayh and Imam Ahmad, according to one report narrated from him. It was also the view favoured by Shaykh Al-Islam Ibn Taymiyah. 
Those who supported this view said: this is in accordance with the principles of Shari`ah. The `Iddah (in the case of divorce) was made three menstrual cycles in order to lengthen the time during which the husband may take back his wife, so that the husband will have enough time to rethink the matter and to be able to take his wife back during this time. 
But in the case of Khul`, the wife is not going to go back and the purpose is simply to establish whether she is pregnant or not, so one menstrual cycle is sufficient. They said: this does not contradict the ruling that the `Iddah for a woman divorced by Talaq is three months. In the case of divorce the `Iddah is the same whether it is a final divorce or a revocable divorce .” ( Zad Al-Ma`ad, 5/196, 197)
 Some scholars said that the `Iddah for a woman who ends her marriage to her husband by Khul` is three menstrual cycles, as in the case of one who is divorced by Talaq , but this was expertly refuted by Imam Ibn Al-Qayyim, who said:
“What indicates that Khul` is not the same as divorce is the fact that in the case of a revocable divorce after consummation of the marriage, there are specific rulings which are not applicable in the case of Khul`.

That the husband has more right to take her back.
When the husband issues a divorce, is it counted as one of three, and after the third it is not permissible for the wife to go back to him until she has been married to another man and that marriage has been consummated.
`Iddah in the case of divorce is three menstrual cycles.

 So the `Iddah in the case of Khul` is what is indicated by the Sunnah, namely one menstrual cycle.
 And Allah knows best.