The ‘iddah of a woman divorced by khula’ and her going back to her husband

If the woman who has been divorced by khula’ is pregnant then
her ‘iddah lasts until she gives birth, according to scholarly consensus.
Al-Mughni, 11/227.

 But if she is not pregnant, the scholars differed concerning
her ‘iddah. Most of the scholars said that she should wait for three
menstrual cycles, because of the general meaning of the aayah
(interpretation of the meaning):

“And divorced women shall wait (as
regards their marriage) for three menstrual periods”

[al-Baqarah 2:228]

 But the correct view is that it is sufficient for a woman
divorced by khula’ to wait for one menstrual cycle, because the Prophet SAWS
(peace and blessings of Allaah be upon him) told the wife of Thaabit ibn
Qays, when she divorced him by khula’, to wait out the ‘iddah for one
menstrual cycle. (Narrated by al-Tirmidhi, 1185; classed as saheeh by
al-Albaani in Saheeh al-Tirmidhi, 946). This hadeeth refers
specifically to khula’ divorce whereas the aayah quoted above speaks of
divorce in general. But if she waits out an ‘iddah of three menstrual cycles
that will be more complete and will be on the safe side, and will avoid an
area of scholarly dispute, as some scholars say that she should wait for
three menstrual cycles, based on the aayah quoted.

Fataawa al-Talaaq by Shaykh Ibn
Baaz, 1/286.

 There is nothing wrong with them remarrying with a new marriage contract. See question no. 10140.