The question at hand is, short of homeschooling (which is not impossible) whether there will ever be a good fit. While she's academically ahead, and getting further ahead every day, she's also impulsive, emotional, and immature around other kids.
We chose not to take our DS10 out to HS. He was very difficult as a kindergartner, tending to melt down, act on impulse, be off track, focus on his own interests, distract the class etc. But we wanted him to learn the skill of interacting with others in school and elsewhere. We wrote tons of behavior goals into his IEP and made sure they were worked on. At 10, he can participate meaningfully at school, has friends, and is doing pretty well overall. We are glad we stuck it out, even though it meant a couple of impossibly hard years early on. Had we homeschooled him, he would probably not have learned all these soft skills that we believe will be necessary for him to hold a job down the road.
Moomin, is your DD in public school? If so, have you started the "educational evaluation" process (called RTI or ETR in most places)? I would bet that if your DD is going to master these participation skills, she'll require management and direct instruction in those skills, as well as sophisticated thinking about how to make it worth her while to participate. (For that job, a behavioral psychologist can be useful.) An IEP is probably necessary in order to get those services deployed in a school setting.
DeeDee
