If you do not want to maintain your asylum anymore, you can do one of two things. You can either transfer maintainership of the asylum to another user or delete the asylum account.
Deleting an asylum is the same procedure as deleting a personal journal [https://www.insanejournal.com/support/faqbrowse.bml?faqid=14]. Simply select your asylum's username from the "Work as user" drop-down menu and click "Switch" before you delete it.