You do not need a lawyer to apply for a resident permit for your daughter. You send an application for a resident permit for the purpose of family reunificiation to your local Ausländerbehörde.
You will have to submit her passport (when it is issued) and a copy of her brith certificate.
The Ausländerbehörde will want to see a document regarding the child's custody arrangements. Either you have a document showing you have sole custody or else, in the case of shared custody, you will have to show them that the other parent has given their consent that she lives with you long-term.