Simply said, you can’t delete your Current TV account at all. You could try to contact them via their Feedback service:
While you request to having your account removed, please complain about the lack of an acount termination service. All sites should have that, no excuses!
Please note that AccountKiller.com is not associated with the business listed on this page. AccountKiller provides easy-to-understand instructions and guides on how to cancel charges and relationships from the list of businesses on our site, we have no affiliation with these businesses.