Answer
(Fatwa: 948/751=L/1430)
The consent of both the husband and wife is necessary for khula to take effect. So, if the husband of your sister first promised to give khula and then denied the same, the khula will not take place. But, if the khula took place orally i.e. wife abandoned her claim of mahr (dower) and the husband in return said that he gave khula or gave talaq, then the khula took place, no matter he signed on paper or not.
Allah (Subhana Wa Ta’ala) knows Best
Darul Ifta,
Darul Uloom Deoband