NepaliBaje,
Your daughter has to be Green Card holder to be eligible to become automatic citizen. I had the same situation in my son's case. In my case, my son got green card when I became green card holder through employment. After I became US citizen, we applied for his passport directly. He didn't have to go through the naturalization process because he was a minor.
I am not sure, my situation does help you to understand your process.

Thanks