What is the ruling of Islamic law concerning how to deal with a brother's wife who screams at and insults the mother, and has gone so far as to insult the younger brother, knowing that he will not respond to her, and what is the correct course of action in such a situation?

Your brother's wife is not permitted to abuse your mother, and you must defend her. Your brother must prevent his wife from abusing his mother. You have the right to retaliate in kind for the insult directed at you, but forgiveness is better. Furthermore, the sin falls upon the one who initiated it, unless the aggrieved party exceeds bounds in their response. Wisdom and careful consideration of matters are necessary to avoid corruption, and patience and forgiveness are superior.