If a uti caused delerium, why does delirium not disappear when uti clears up?

Systemic infection: Will often trigger acute delirium in an elderly patient. And you are correct, the infection is likely to improve faster than the mental status change. Be patient.

Life of its own : Once delirium starts it often is a secondary process apart from inciting event. Also, we often find a UTI or other problem when we are looking to explain acute increase in confusion. Just because there is a uti, does not mean it alone caused the confusion. This is why prevention, early recognition and treatment for delirium are so important.