My brother tried to do something similar, but the birth father wouldn't relinquish his right to parent so it never happened.
It is likely that your daughter's father will have to allow your husband to adopt your daughter. In doing so, legal guardianship transfers to your husband. In the meantime, as would have been the case in my brother's situation, the birth father no longer is responsible for child support (since he is no longer the legal guardian), so if you've been receiving child support from the birth father that is likely to cease (although my brother's case was in a different state so different rules may apply).
Your best bet is to contact an attorney who specializes in this sort of thing. You will be transferring legal guardianship from one adult to another, so there will be a set of specific steps that will need to be followed.
As far as your daughter is concerned: ask her what she would like to do. It is possible perhaps that during this name change process she could keep her original last name but as a middle name, such as going from Jane Anne Doe to Jane Anne Doe Rogers.