What do you want to pay her? If it concerns the equity she contributed, it's okay, but if it concerns a value that was financed anyway, in my opinion, everything has been settled.
Well, she probably owns half of the property including the house and is likely jointly liable for the loan. You have to clearly separate that, and in the end she is selling him her half of the house and thereby redeeming "her" half of the loan. The bank also has to be involved, because their securities decrease since only one debtor remains.
Isn't a gift or something similar possible? You can also simply set the sale price low. Who is forcing you to actually sell the half of the house at the market price?
That is clear; in my opinion, it is a question of weighing which is higher. Real estate transfer tax or gift tax; after all, it is not the original purchase price that counts, but a portion "x".