In Michigan, contracts for sale of real estate and for leases longer than a year must be in writing, or are not enforceable. There are some ways around this regarding the term which has been performed, changes of position in reliance, etc., but in general, with the facts you have provided (i.e., it has already been more than a year, and nothing exceptional which would seem to warrant some form of limited equitable exception), it would seem the tenant is not obligated for remaining rent after moving out. Like another responding attorney indicated, the tenant may be responsible for rent as long as it stays, and if paying rent on a monthly basis, for giving 30 days notice of moving, but aside from liability for causing any physical damage beyond normal wear and tear, there should be no further rent liability. If a security deposit has been paid, there may even be recourse for the tenant to get that back.
An unsigned contract is not a contract. It is possible that the actions of the parties May evidence a contract. Most likely a judge would find that you have a month to month tenancy.
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I agree with my colleague, absent some other conduct, an unsigned contract is not generally binding. I also agree that what you most likely have is a month-to-month. For advice specific to your circumstances, sit with an attorney for an hour or two and get his or her opinion. Good luck.
Answer given for general advice and is not a legal opinion, which would require an analysis of the facts and circumstances as well as the applicable law and regulations.