Contested Step Parent Adoption

I have been raising my step daughter for 3 years. My fiancee, her father, and I will be married in June. I would like to adopt her before then. Her biological mother will say no for sure and fight 'til the death even though if my step daughter stood right in front of her, she wouldn't recognize her. My fiancee has had sole custody my step daughter since she was 10 days old. Her biological mother has had visitation of which she has defaulted. She does not and has never paid child support. She is also bi-polar, suffers from several mental ailments and has been in and out of mental institutions for the past 5 years. Does anyone know the laws of NYC of hand? Has anyone in NYC gone through something similar? We aren't well off and cannot get into an expensive legal battle. Thanks.