What is the way to get rid of intense jealousy over one's husband, especially concerning the actions of his brother's wife who fulfills his requests at home, and the desire to take care of the husband's affairs instead of her?

Jealousy is commendable if it is in its proper place, and Islamic law has set boundaries for the interaction of men with non-mahram (definition) women, and has warned against leniency in this regard with the husband's relatives. The wife should fulfill her husband's requests and not fall short in serving him so as not to leave room for his brother's wife to serve him. She should gently tell her husband to ask her directly for what he wants, while being keen to obey him in what is good, be a good wife, and cooperate in obedience to Allah.