My ex girlfriend is currently incarcerated and in exchange of me dropping the domestic charges against her she will sign over all legal rights of my daughter to me? What legal documents are required to do this and what steps will need to be taken?

You have to go to Court and to ask for sole legal and physical custody over the child.
You may make an arrangement.
Have a look here
Free Legal Forms (http://www.freelegalforms.net/FreeLegalForms.cfm?criteria=custody)

It is unlikely that she will be allowed to sign over her rights. There is this myth that a parent can just sign away their rights to a child, but its only a myth. Only a court can terminate parental rights and courts are very reluctant to do so.

You mentioned domestic charges, if she is violent and can be proven a danger to the children a court might issue a TPR.

On the other hand, under the circumstances, you should be able to get sole legal and physical custody, especially if she is willing to agree to it. But this opens the possibility that, at some future time, she can apply for visitation and partial custody.

I would suggest consulting an attorney to prepare the paperwork.

First she can sign custody over to you, but she will still have the right to ask for visitation after she gets out. Or even refile for custody again after she gets out.

Also if you suggest this without an attorney, that is considered blackmail and illegal, Your attorney can ask for various legal deals.

Next you don't have the ability to "drop" charges, only the DA can do that,