What is it that we need to be sure to get in writing?
Whatever you negotiate with the landlord that is different from what is written in the lease, or in the law if the lease says nothing about what happens if the tenant moves out before the end of the lease.
We were thinking of asking for an insecentive for us to get out faster, but we are going as fast as we can now.
Sounds like you have your hands full just dealing with this and don't have the time or energy to try to exploit any money-making opportunities the situation might offer (which might not exist anyway). The main thing to do is to make sure the landlord won't try to charge you rent after you turn the apartment over to them.
In a normal rental market, the landlord WOULD charge the tenant rent for the period before a new tenant starts paying, and it would be fair and reasonable for the landlord to do so. But I believe the NYC rent-control and rent-stabilization laws mean the landlord can remodel the apartment after you leave and then charge a much higher rent rate to the new tenant, so the sooner the landlord can start that process the happier he/she/it will be.
The thing is, if you just leave and hand over the keys, an unscrupulous landlord could quite possibly charge your aunt rent during the time the apartment is being remodeled AND then charge the new tenant the higher rent afterward. Maybe if you refused to pay and it went to court you would win, but you really don't need THAT hassle in your life either. Much better to get the landlord to sign an agreement to charge your aunt rent only through the last day you actually keep possession (they may need you to tell them which day that is at least a little in advance) AND to provide you with a written list of any damage they intend to charge your aunt for BEFORE you have to hand over the keys (so you can get the repairs done yourself if necessary). Hopefully what will happen is they will be in a hurry to start their remodel so they won't try to charge your aunt for damage to things in the apartment that they plan to replace anyway.