I found out she was cheating on me with multiple other men

I need some information about father's rights. Before I get started I'll give you some background. About a year ago now I started seeing a girl named Taylor. We were doing good in a relationship and she told me she was pregnant. Which was fine I was okay with having a kid. But a few days after that I found out she was cheating on me with multiple other men. None of us knew who the father of the child was now so I ended things with her. About a month ago we did a DNA test and it shows that I am the father. After that she asked that I take her back in a relationship and I said no.

So now she is trying to manipulate me and control me using our daughter ______ as a bargaining chip. _____ is still in the hospital as she was born with gastroschisis. Now her mother tells me when I can and can't visit and has me on the no visitor list and the nurses won't tell me any info when I call because mother says no. Also my name is not on the birth certificate yet due to the weird way things all happened. How can I get my name on the birth certificate now?