I think you just give them your info as well. In my case my husband was on my insurance so we didn't have to deal with it. You are correct that for the first 30 days you do not need to add her to your policy, but it often takes time for them to process any plan changes so you probably want to get started on it asap. You will probably need to order a copy of the birth cert before she can be added.
We discovered that my work has a policy that any new babies must be added within 60 days of birth, and of course are covered automatically for the first 30 days. It left a 30 day window though where she wasn't technically covered. Our peds office usually sees newborns at 1wk, 2wk, 4wk and 8wk, so during that window we generally wouldn't have needed coverage. However, both of my kids were born with troubles that required specialist care during that window. I knew their insurance would be back-dated to cover the 30 day gap, but the providers all had to list us as self-pay anyway and made us sign consent forms to be self-pay. Thankfully it all worked out but it was a ton of added stress.
You mentioned your daughter is in the NICU. I'm not sure which state you are in but in CO if your child is in the NICU for 30 days they are automatically enrolled in CHIP, which is medicaid for children, regardless of income levels. Then they are covered under that for the first year. If you are in there that long you might end up in a similar situation.