Is it permissible for me to give charity from old usurious money that my mother took from the bank, without her knowledge, and is her knowledge a condition for Allah to accept [the charity] and forgive her sin?

If the bank statements prove that the mother received interest from old deposits, then she must repent and disburse the equivalent amount of these interests for charitable purposes and the general welfare of Muslims. If she refuses after it is proven that she received the interest, she is sinful. If the daughter pays the money on her behalf, the financial obligation is lifted from the mother, but the sin of non-compliance remains upon her, and she might be excused due to her ignorance. The acceptance of repentance is entrusted to Allah.