Does a wife's negligence in performing her religious duties, especially delaying prayer, despite the husband's repeated admonition, obligate the husband to leave her as she is without sin on his part, or does she alone bear responsibility for that before Allah?

Allah has granted the husband the right of qawamah (guardianship/headship) over his wife, which involves overseeing both worldly and religious affairs. This includes encouraging her to perform her religious duties and avoid prohibitions. However, this should be done with gentleness and wisdom. Therefore, you should admonish your wife with gentleness and patience, and strive to help her strengthen her connection with her Lord. You are not permitted to abandon advising and teaching her, nor should you coerce her. Indeed, the heart of a believer should expand with mercy and compassion for those who err.