Early termination of 11 month tenancy agreement

Hello,

Where do I stand legally if a tenant has terminated their 11 month tenancy agreement early given only a months' notice and using their deposit as payment for the last month? The contract states that both parties have to give 2 months' notice if they terminate early. Are they only liable for the 2 months' notice period in terms of rent or for the full rental period?

Should I accept their termination in writing and hope they leave on the said date, or should I insist on the two months' payments they will owe running the risk they pay and stay for the full duration of the contract or at least the next couple of months? (I have had numerous problems with these tenants and prefer them to leave asap).

Kind regards

Stimopo