Is it permissible for a brother to pay off his sister's debt from his zakat money, knowing that she is employed, divorced, has two daughters, and bought a house with a loan to live in?

It is not permissible for the brother to take his debt from his sister's Zakat, nor is it permissible for her to give him Zakat on the condition that he repays the debt. Forgiving a debt from Zakat is not permissible according to the majority of scholars. Similarly, paying Zakat on the condition that it is returned as a debt is invalid and does not fulfill the obligation of Zakat.