EM - you sound as if you are in a very similar situation to mine this time last year. My DS had struggled with school on and off (a lot of sickness, just like your DD), but in Year 6 he became seriously depressed. He didn't get on with his teacher and was being bullied by his peers, so every school day was awful. I decided to deregister him, but the school suggested they kept him on their register so that he could go back if he wanted. Our GP signed him off sick and we started home education. We had several really happy months. It was absolutely the right thing to do.
DS did his SATs six months later (we chose to do them - you certainly don't have to put your DD through them if you don't want to) and, amazingly to me, had gone up a level in English, Maths and Science (from Level 3 at school to Level 4 in the SATs). I was astonished because I didn't feel I was doing enough. But, of course, DS was getting one-to-one attention every day. He played BBC Bitesize on his computer and we went through some workbooks together. And we talked and just had a really happy time together.
You say your other child has Asperger's and you wonder if your DD has too. I presume you have had her assessed and she doesn't reach the diagnostic criteria? Would it be possible to seek another opinion? I say this because it may be that she needs help at school that she isn't getting. In my son's case, I asked for a third opinion (he had twice been assessed for AS without getting a diagnosis) and we finally saw a psychiatrist, who diagnosed him (as we had long suspected) with Tourettes and OCD. He is now on a low dose of Prozac and is having CBT to help with his OCD, which is far more troubling than the Tourettes. The Prozac has taken the edge off his anxiety, which was crippling for him even when he was at home all the time.
In September, my son started at secondary school. He had numerous visits to the new school first, with an Education Outreach Worker who was responsible for reintegrating him into school. He is in a small class with teachers who understand his difficulties and never mind if he needs to leave the class to calm himself down. He loves the lessons, which he finds far more interesting than those at his primary school, and he is thriving socially. He loves his newfound independence and his confidence is growing daily.
Year 6 is a good year to miss because it is usually just spent revising for the SATs. It is very boring and difficult for many of the children. Missing that year gave both my son and me a useful breathing space to reassess what we should do next.
Try not to worry about what other people say. I pointed out to my critics that DS was receiving one-to-one education and, luckily for me, when he achieved good SATs results that proved my point and shut them up. I know people in the family criticised me for "pandering" to DS and said I should just force him to go to school, but they really didn't understand how serious the situation was. People are always so full of opinions about things they don't know anything about, aren't they?
Best of luck. It sounds as if you are doing the right thing for your daughter.