I'm not a lawyer, so this isn't legal advice. For that, you need a lawyer, however . . .
Even if the lease contains both elements (an expiration of 3/31/2010) and a statement that a 60 day notice is required, it can depend on the context of the two provisions.
For example--and, again, I'm not a lawyer--if the paragraph read something like: "Lease to expire on March 31, 2010, but will automatically be extended unless notice to terminate is provided in writing prior to January 31. Otherwise, lease may be terminated upon 60 day's notice." Then you'd be stuck; the lease would be clear about an automatic renewal and the required 60 day's notice.
On the other hand--again, I'm not a lawyer--if the paragraph read "Lease to expire on March 31, 2010." But another provision of the lease, dealing with early termination, said, "In the event of termination prior to expiration of the lease, tenant is required to provide 60 day's notice," then you'd have a very strong case.
On the other hand (we're up to three hands now!), if the paragraph in question read: "Rent payment shall be due for a minimum of 60 days following written notification by tenant of intent to vacate premises, regardless of the scheduled expiration of the lease," then you'd owe the money.
So--to summarize--it really depends on how and where that 60 day notice is written in the lease.
It appears that you didn't give proper notice however, I don't think that they can obligate you to a new payment without a new lease. If you did not give the proper notice the lease will convert what is called a tenancy at will, but I am almost certain that the existing terms must stay without any modifications. I would let them know that you would really like to leave, and offer to give them something to let you go. Failing that I would let them know that if they are going to enforce the tenancy at will then they must enforce the entire contract including the rental amount.
Does anyone else out there have any experience with this. I know that you can change terms and ask a person to sign a new lease but I am almost certain that you cannot change the terms and try to enforce them on a tenancy at will. It meets the common sense test at least. What would stop a landlord from changing the price to something outrageous and then trying to enforce their rights in a court of law.
This is not legal advice - should you desire that you should call an attorney.