What is the way to marry a girl whose father refused to let me marry her, after he heard from his wife an impulsive remark from my mother that she had beaten one of her sons' wives, and with my knowledge of the girl's desire, and with the refusal of all her family?

If the girl is religious and of good character, then continue trying to convince her father. You must supplicate and seek help from Allah. You can also enlist some of her father's close associates to influence him. If he agrees, then that is well and good. But if he insists on refusing, then leave her and look for someone else. Remember His (Allah's) saying: "And it may be that you dislike a thing which is good for you and that you like a thing which is bad for you. Allah knows but you do not know," and His saying: "But perhaps you dislike a thing and Allah makes therein much good." And sever all ties with her, for she is a stranger to you. It is not permissible for a guardian to prevent his ward from marrying a suitable man if she is pleased with him.