What is the legitimate Islamic solution for dealing with a wife who insults and slanders her husband's mother, and presents a different image from the one he knew her by, keeping in mind that he does not wish for the destruction of homes or the instigation of problems?

Spying on one's wife is forbidden unless there are strong suspicions. What your wife said about your mother, unjustly, is against Islamic law and morals, but it does not warrant the destruction of the marital relationship. One should not judge a wife based on a single mistake; rather, all her aspects should be considered, and her good deeds should be weighed against her bad ones, as Allah Almighty says: "And live with them in kindness. For if you dislike them - perhaps you dislike a thing and Allah makes therein much good." [Quran 4:19]. And the Prophet, peace and blessings be upon him, said: "No believing man should hate a believing woman; if he dislikes one of her characteristics, he will be pleased with another." You should gently and wisely advise your wife to be keen on good character and to stay away from oppressing others, backbiting, and slander. Explain to her that the kindness of each spouse to the other's family is a sign of complete good companionship.