Emergence of khula is product of judicial activism in Pakistan. This new kind of divorce was created in case of Balqees-fatima v najam-ul-ikram. However,now it is so widely accepted, that since, this judicial khula is incorporated in pakistan, Bangladesh and Egypt.
Source of khula is not anything else, but, Quran and Hadith, itself. It draws it validity from surah 2:229, 4:35 and Jameela-Hadith. As surah 2;229 says
" But if you fear that they will not keep [within] the limits of Allah , then there is no blame upon either of them concerning that by which she ransoms herself."
So here learned judge interpretted "you" as judicial officers.But, the important condition to this right is that, judicial officer should be satisfied that they can not keep with in the limits of Allah. So this right is not absolute.
Though, this interpretation might not sit well. But the reality is that it is true interpretation according to Jameela-Hadith. Here judge argues that Prophet(S.A.W) was acting in his judicial capacity.
"Wife of Thabit ibn Qais(RA) came to Nabi (Sallallaahu Alayhi Wasallam) and said(Radhiallaahu Anhu), I do not fault his character nor his Deen (religion),but I fear being ungrateful in Islam. I want to return his garden to him in exchange of Talaaq.Nabi (Sallallaahu Alayhi Wasallam) said (to her husband), ‘She wishes to return your garden to you(in exchange for Talaaq and pronounce divorce.
Hence this right confirms the validity of khula in Islam. Even, not only this but surah 4:35 strongly supports this right as under this surah there is power to reconcile by appointment of two arbitrators, one from each party. And than the decession of those arbitrators are binding.
Now, there seems to exist no doubt on valid basis of khula in Islam. However the debate in what circumstances is grant of khula valid still ongoing AbdulRahim v shahida.