Q: How do I delete my alternate apple ID?

You can't, but you can stop using it.

But here is my problem.  Until Family Sharing came along, our family shared my Apple ID. My daughter put in the @me cloud account which I didn't know about.  Now that we are trying to use Family Sharing, I cannot send her an invite because it says she is already in a family... I am so confused!  I am assuming it is because of her adding herself as the alternate Apple ID.

I have the same problem with my child.  Did you solve it or just change her id to add her to family sharing?

Thanks