GA Tenant telling me they are breaking the lease - what do I do?

[Taylor Larkin]

Newbie here.  I purchased my first rental on August 30th, a duplex that already had tenants.  Both tenants had been there for more than 5 years so I wrongfully assumed both were stable.  As soon as I took possession, both tenants signed a new 12 month lease with me and both paid rent in Sept.  One of the tenants informed me on 9/21 that she intended to move back home to be with her ailing mother and father.  She has no intention of following the stipulations in the lease for breaking it (45 day notice, rent due during the notice, plus an additional months rent).  I'm ok with making some concessions and working with her outside the specifics in the lease but I want to get them in writing from her. 

Can we just do a move-out agreement that lays out what we are doing differently?  I just want something in writing.  And if I do that, can I try to rent the unit out immediately?  Any thoughts or comments would be appreciated.