Someone correct me if I'm wrong, but I thought autism always comes with sensory processing disorder?
The difference between aspergers and autism was learning language. If they speak well they were likely to be diagnosed with aspergers and not autism, regardless of how they learn and how independent they are. That's all. Regardless, aspergers is not in the DSM anymore which means your daughter falls under the autism umbrella now.
I had a lot of difficulty in school but I think a lot of that was because I wasn't liked by my teachers or the other students. I was usually ignored whenever I had a question about something, so I never fully understood whatever I was supposed to be learning. That led to me getting bad grades, which was frustrating and eventually I stopped doing homework and schoolwork entirely because there was no point if I was going to get bad grades anyways, which just made my teachers like me less. I was really quiet but by about 2nd grade I was a very angry child, just too frustrated for too long, that's when behavioral problems started. I was also a foster kid so there were other problems at home, etc. I hadn't been diagnosed with autism so I didn't get any kind of help from my school in the areas I needed it in, I was just labelled a bad kid.
Focusing on things that aren't interesting is difficult even now, because my head is so full of other stuff all the time. Keep working with your daughter on her writing and phonics, make it more interesting, maybe have her write names of penguins or something like that. Ask her questions about how she prefers to learn and really listen to her.