Am I obligated to provide for my divorced wife for two years that she spent with her family after I hit her for her wrongdoing and her brother took her without my permission, knowing that she is disobedient and leaves my house without my permission, and my children are with me?

It is not permissible for a wife to leave her husband's house without his permission. If she does so, she is considered disobedient (nashiz), and no one is allowed to take her or detain her from him. If she leaves disobediently, her maintenance for that period is forfeited, unless she is pregnant, in which case some scholars deem her entitled to maintenance. It must be noted that striking a wife is only permissible for disciplinary purposes, within شرعي (Sharia-compliant) guidelines, meaning it should be light and not cause harm.