Hey this answer may be too late seeing that your post was from 2 years ago. anyways how is that problem of yours with your daughter going? Did you somehow find a way to make things okay?
My advice would just be that I think first and foremost if you want to tell a child something you have to know it first. As you mentioned when you were young you kind of struggled with the same problem. How did you resolve that? Have you really resolved that for yourself?
I'm of mixed race and I've never had this problem because both my parents are very proud of their respectful origins so this way I have never felt that one heritage was less desirable than the other. Maybe to help your daughter you need to show her how proud you yourself are to be black, and show her what it means to be black. Teach her about whatever culture it is that afro-americans have, let her experience how it is to be in a black family, so get her to interact with her grandma and aunts, cousins. Let her feel what it means to be black and surely she will feel proud of her black side. But at the same time also honor her white heritage.. a balance of both.
And let her know that being more than one race, she has more diversity, more beauty. She is two worlds in one.