@Eddie Fleckenstein Thanks for the additional info. You could be in for a bit of a fight if she doesn't leave on her own but that's just how it always is.
Not knowing the additional names could be an issue down the road but what can you do.
When you purchased the place did you have the seller/landlord give you letters of indemnification signed by landlord and tenant? Did he transfer any security deposits and last months rent (or sign off that there are none) That all could come into play if it gets to court.
Did you send letters to the tenants stating you are now the owner and rent should be paid to you etc? How do you know she didn't meet the application requirements? Did you give her an application to complete and then declined it? If you gave her an application you should have asked for everyone living in the apartment etc.
Perhaps what you should have done is given BOTH tenants notice of new ownership, advised their leases are expiring 8/31 and in order to possibly renew they would have to complete a new application and meet the new criteria (which you have written down and applied equally to both tenants) THEN advise her she didn't meet the criteria and therefore lease not being renewed. But it is a little too late for that so you will have to run with what you have.
So once she gets the notice she will be calling you. Be very careful what you say, and make your own notes afterwards. Just keep it simple - the tenancy is not being renewed, please move out by the end of September. If you have already told her she doesn't meet your criteria, don't get into it with her about how or why etc! She may counter that she needs more time to find a place, save up the money etc. DON"T agree to more time up front. If she doesn't move, file the court action. If in court she says she needs more time, then it can be written up as an agreement in court. That way if she doesn't move you can get the Execution rather then starting the court process at that time.
One more thought, since this is Chelsea, there are various programs in place that will help tenants with various costs, including moving costs (and first, last security at a new place) You may get a call from Legal Aid as well that they are helping her.