I'm glad you have brought up the possibility of Aspergers here Sar53. I see a lot of parallels in your post, with my DD and DGS1. My DGS has Asperger syndrome, he was diagnosed age 5. He never had any friends at school, only ever managed to be invited to one birthday party, and was miserable at school. My DD had only one real friend at school although she managed very well with the work. She used to say similar things to me as your GD does, and was often really horrible to her little brother, she has never had a formal diagnosis but we suspect she too has Aspergers.
On a positive note, my DD went to University, got a degree in Anthropology, and is now a wonderful mother to four lovely, if difficult children. A diagnosis is just a label, with really good parental support, and your help she will turn out very well.
Good luck for the future.