How can a person who regrets betraying his wife and has repented to Allah convince his wife of his repentance, and is his repentance considered sound and complete?

All praise is due to Allah, who has blessed you with repentance. Your repentance is valid as long as it fulfills its conditions, and through it, you return as if you had never sinned, for the Prophet, peace and blessings be upon him, said: "The one who repents from a sin is like one who has no sin." And increase in good deeds and supererogatory acts, for indeed, good deeds efface evil deeds. Inform your wife that you have repented to Allah and that the one who repents from a sin is like one who has no sin.