You need an EHCP for the council to even consider paying.
And even then the assessments done would have to advise that your dd needs support that cannot be offered in a mainstream school, except by costly extra teaching support, which might make it cheaper for the council to pay for a dyslexia school placement.
In London there are two schools that spring to mind: The Moate School and Fairley House. Fairley House seems to be for students with no behaviourial issues apart from a diagnosis of dyslexia.
Outside London there is More House in Frensham I think. A lot of children at these schools are funded by EHCPs, but I suspect that in order to even get to that stage the parents have to prove a serious disadvantage in their children's current progress. Just one assessment is not going to get you an EHCP. But you can certainly apply for one. Contact IPSEA for further advice.
Son has EHCP for dyslexia dyspraxia and asd and is not in a specialist school. He is doing really well at the right mainstream comprehensive and is scribed for exams atm. He gained excellent gsce results and is now doing A levels in humanities. He doesn't have a 1:1 nor did he, but he had some specialist teaching input and some smaller classes. His school has a good reputation for SEN support but they are now feeling the cuts with the new funding system for EHCPs which adversely affects them, where before the money they received from the council could be pooled more efficiently than now, where there is shortfall..
I recommend speaking to the SEN in all the secondary schools and finding out what they offer in the way of specialist support for dyslexic pupils...could be typing homework, homework only in core subjects, touch typing clubs, pastoral support in other ways to increase confidence (ie extra curricular opportunities)
The working memory and processing would be the biggest issue for my child in terms of taking in instructions or understanding what was asked of them in classwork, far more than just reading ability or spelling. whatever you decide about the EHCP (and some children get no particular benefit financially from cash strapped councils despite having them) just knowing what your child's difficulty is fore arms you to seek the advice and support of the teaching staff and support your child.
Ds as I say is doing very very well and works hard but I had to take him out of school at the end of year 7 partly because I didn't really know how to tackle the problem and was trying to fit him in to school system. 2 years out and more 1 to 1 input at home and he went back to school with much less stress, ready and able to learn in classroom environment. I don't think if he had just perservered and floundered at his original school the story would have been the same. I needed to know what strategies would help him, in detail.