let along a single mom with a five-year-older. My suggestion to you, from a practical point of view, is to play nice with the tenant and even offer her some money to move out (yes, forget about the rent). Of course, it is of your right to evict her for holdover. However, almost all the landlord tenant court judges in the City are very sympathetic towards the tenant. So, you would be lucky to get her out within 6 months. Check this the following official landlord guide for your reference. If you do decide to do it the hard way, make sure you retain an experienced landlord-tenant attorney, as the burden to serve the correct notice in a way proscibed by the court is not something a layman want to take on.
As for your coop board, please try to communicate with them. I cannot think of a reason that they would give you a hard time if you have a viable plan to get the tenant out.
http://www.courts.state.ny.us/courts/nyc/housing/pdfs/Landlordbooklet.pdf