The answer you seek is not as to the existence of the lease. As it was signed, it is in effect. The issue is what will it take for you to establish the present right of possession? And then to collect the deposit and rent that will accumulate until that can be settled? For example, in December this tenant could show up and demand possession!. So depending on the terms of the lease, you might claim the full year's rent! You also need to both claim damages until you can have the lease terminated and to find a new tenant. Go back to your attorney for help.
In Ohio, lease agreements are treated like any other contract. A tenant who signs a lease agreement is obligated to honor the terms of the lease. If the lease agreement includes specific provisions for early termination then you are entitled to the remedy set forth in the lease. However, as with any claim for breach of contract, you have a duty to mitigate your damages. The duty to mitigate, in this case, would require you to undertake some effort to re-rent the property. Depending on the provisions in the lease agreement, the tenant may be liable for the lost rent while the home is vacant and the costs reasonably and necessarily incurred to re-rent the home.
A lease agreement is a binding contract between you as the landlord and the tenant. You are owed the total rent due under the full term of the lease, until the tenant complies with any lease terms that allow the tenant to terminate the lease early before the term of the lease expires. But, also, you do have the duty to mitigate (minimize) your economic damages by making reasonable efforts to lease the premises to another tenant. Bottomline, yes, they owe you at a minimum the rent payments for a reasonable number of months until you can secure a substitute tenant, probably at least 2 or 3 months. To collect, you can sue them yourself without an attorney in the Hamilton County Municipal Court Small Claims Division. Go on-line, google the court's name, and the site should direct you to the necessary forms to complete and instructions for filing. Of course, your lawsuit only gets you a judgment, not money in your pocket, which is another legal proceeding if they don't pay the judgment voluntarily. Your offer to accept only one month's rent to cancel the lease agreement is very reasonable. (Next time, ask for more than your bottomline to give yourself a little room to negotiate.) You might go back to the tenant and tell him or her that the tenant has one week to accept your settlement offer and pay the full month rent by cash or cashiers' check or the offer is automatically withdrawn, and you will seek the full amount the law allows. That might shake the one month out of them. Good Luck!