Is it permissible for a brother to give his sister from his Zakat money and from bank interest to help her purchase a house with a large debt, knowing that her salary is not fixed?

There is no harm in a brother giving his zakat to his poor sister who does not possess what would cover her debt, unless she is someone whose maintenance is obligatory upon him. It is also permissible for the brother to give the forbidden usurious interest to his sister to pay off her debt, and he must repent from dealing with usury.