what is the legal time frame I have to give in NYS for someone w/ children to evict?

This tenant is a piece of work. I am new to the whole renting out process and is only doing it because I need to pay the mortgage one way or another. Its a mess my parents left my husband and I with. Anyhow, I asked and was advised that since this girl has been giving me one excuse after the next and only her first month in the house she is late on rent (16 days now but she thinks its only tech 10 days as if that makes a diff) and doesnt look like she is planning on paying, that I can serve her with an eviction notice. I looked up a form on line and and used it and gave her 3 day to pay or leave clock starting today. She actually has the guts to txt me today… she said my notice was illegal , renters righst ppl with kids have 30 days to evict and she wants her security deposit back and I will be hearing from her attorney because she wants to go to court.. I said NP, plz have ur attorney call me today .. so now she will cost me $500 because I have to hire an attorney to evict her when I know if she cant afford to pay the rent she cant get an attorney … I would be so embaressed if I were her I cant believe she actually told me that .. I cant believe there are ppl out there like that . This place is costing me more than when it was empty. Is this true about ppl with children ? and also I am really not doing this to be mean , but I am behind on bills because I was depending on that rent money and didnt budget for things otherwise, and I cant have someone living there for free … so how long would this process normally take, while I continue providing a place for her to live and heating it up for her?
That’s an awesome answer and great help. TY very much