I go along with Froggy and Mommy. You signed the lease. The lease clearly (even if not clear to you) states that 60 days notice is required to terminate the lease. You are therefore responsible for rental for 60 days from notice. The only thing that would change that is if the landlord re rented the unt before the 60 days.
You can nitpick the letter of the law, but Mommy is correct. Housing court is more informal and looks more at intent than the letter of the law.