These sorts of threads always descend into a 'step mum resents step child' narrative.
Big that's not really what's going on.
What step mums resent is the pressure and expectation to be a carbon copy substitute for the step child's mother.
A: that is impossible both in terms of biological reality and in terms of family relationship dynamics
B: no one ever asks the step mum how she feels about it, it's just assumed that she will take on this responsibility without having any regards for where she would prefer to draw her own boundaries. As is her right, by the way.
It is a massive ask of anyone to love another person's child like your own child, when that child will never love you like their own mother. It makes for a really awkward, one way, pressurised, guilt-laden relationship. Horrible for all parties. Especially when the mother is let off the hook for awful behaviour while the step mother is held to unreasonably high and exacting standards.
It's obviously awful that the girl's mother has abandoned her. But the solution is not to install the OP as a replacement. It is the child's father's responsibility to try and manage the situation and the child's grief and upset about her mother, with the background support of the OP.
The issue will not be resolved by the dad simply saying 'it's okay DD, you can have this mummy instead'. It will just be confusing and uncomfortable for both the child and OP.