This scenario is unfortunately fairly common. Since your roommate is a month to month tenant, the police were correct that you must serve her with a 30 day notice of termination. If she doesn't leave at the end of that 30 day period, you'll need to go to housing court to file a Holdover proceeding against her. It's best to hire a lawyer to help you, but you can also use the court's website here: http://www.nycourts.gov/courts/nyc/housing/roommate_termination.shtml#general
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Retain a landlord tenant lawyer and do it right. If you illegally evict her, then you may face treble damages.
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Dear New York Tenant:
The three (3) day notice is a Rent Demand. Should your roommate not pay the "rent" you may commence a nonpayment proceeding. Likely, that course of action would be more difficult than "terminating" the tenancy and commencing a holdover proceeding.
A tenant does not depart in the event of a default in a rent demand. The proper form for the written rent demand requires that the tenant either pay or surrender within three days, but must provide that in the event of a default in paying or in moving out, that a summary proceeding would be brought against the tenant.
A termination notice must end the tenancy on a specific date to be proper. Since the month to month began on the first day of a month, the termination notice must be written and served at least thirty days before the end of the next end of a month. If properly served (and that means by someone other than you) in May 2013, the month to month termination notice would specify June 30, 2013 as the end date of the tenancy.
Once the tenancy is terminated be very careful about accepting rent, although in your situation, it is not likely that you will ever be paid again.
Try to use an attorney as the extra effort will spare you the anxiety of wondering what to do and whether it was done correctly.
Read more about the process for a roommate eviction at:
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.