I'm so sorry you are going through this. It's hard enough leaving your child with someone, and even harder if you know you're child is miserable.
Perhaps try a new nanny? Maybe there is something your daughter doesn't like about her. We went through 4 babysitters before we found one that I liked (and trusted) and that my son liked. But I found a few things to be true...he likes soft spoken over loud or even normal voice tones, attractive over unattractive, and will not tolerate anyone who uses perfume. And actually, this was all his prefenerce as we had a woman with us whom I liked, but my son didn't.
Also, can you allow for more of a period of adjustment? Maybe pay your nanny overtime to tag along with you on errands, stay for dinner, etc. for a few weeks so your daughter comes to see her as part of the family so to speak? It's hard for little guys to be away from mommy and probably scary to be left with a stranger.
My nanny is more of an extra set of hands for me as I'm a SAHM so more often than not all three of us are out together doing things and if I have to leave him alone with her, for say a doctor appointment or something he can't come to, he has no peoblem staying with her. I think he just things she's part of life.
Godo luck!