From the information you’ve provided in your letter, it’s probably going to be impossible for you to secure legal status here for your birth daughter, even though you are a U.S. citizen and the two of you have resumed contact after a presumably long period apart.
When your daughter was adopted at birth, at that time you would have surrendered all legal rights to her. Her adoptive parents are therefore her legal parents under the eyes of the law, and that includes U.S. immigration law.
The exceptions to this are very limited. If the adoption occurred under fraud or duress you could fight its legality in an Irish court, but this would be extremely difficult, time consuming and costly to prove at this late stage. And if your daughter continues to maintain a good relationship with her adoptive parents, that’s a path you surely would not want to pursue.
The other exception would occur if your daughter was orphaned or abandoned by her adoptive parents. If this very unlikely scenario was the case, and if she is under 16 years of age, you could then move to adopt her. Given that you say she wants to live and work in the U.S. it seems reasonable to assume she’s over 16, so this option would not be at all feasible.
U.S. law is very particular about how it deals with adoptions. A U.S. citizen parent can petition for legal status for an adopted child, provided that the child was adopted prior to turning 16, and has been living in the legal custody of the parent for at least two years.
Also, a U.S. citizen who became a citizen through adoption cannot sponsor a foreign-born “natural” parent for legal status here. In other words, if your daughter had been adopted by American parents and became a U.S. citizen, she would not be able to act on your behalf for legal status here if you had remained in Ireland.
What options does your daughter have to achieve her goal of living and working here? You don’t provide information about her background so it’s hard to say for sure, but if she has a college degree and a skilled employment background she might be able to secure a temporary work visa.
If she’s still studying the option exists for her to pursue her studies in the U.S., which would entitle her to an education-based visa provided she goes through the proper channels.
Other than that she can apply for the annual DV-1 green card lottery, but winning one of the 50,000 available green cards is next to impossible given that millions from around the world apply for them.
You should speak with an immigration expert about her particulars if she’s intent on making a life here. As you’ve written from a New York address, you could first try speaking with a counselor at the Emerald Isle Immigration Center in Queens, 718-478-5502, or the Aisling Irish Center in Yonkers, 914-237-5121.