What is Khula
Khula (annulment of marriage) is in fact divorce on demand for separation by the wife by offering some money or foregoing her monetary rights. For Khula, the consent and agreement of both the husband and the wife is necessary. If the wife wants a Khula, she can offer money to the husband and demand it from the husband, but there is no Khula without the husband's consent. Taking Khula without the knowledge of the husband has no meaning in it. Thus, a wife cannot go the Qazi and take Khula without the husband's knowledge. As given in Tabyeen Ul Haqaaiq, Babul Khula, Vol. 3, Pg. No. 189.
When the husband consents to give the Khula, then it is affected. In this case, a Talaaq-e-Baain (an irrevocable divorce) is carried out between the wife and husband, as given in Fatawa A'alamgiri, Vol. 1, Pg. No. 488.
If they want to reunite out of their own free will, then within Iddah (the mandatory waiting period) or after it, they can remarry with a new Mehr (alimony) before 2 witnesses. As Khula is like an irrevocable divorce, there is no need for Halala after it.