Move quickly. At 16 your daughter will be at the end of her statutory education and many LAs (including mine) will state that they have no responsibility to fund post-16 education.
I was in a very similar situation to you with LA and SS both refusing to acknowledge that DS needed residential placement to enable him to leave Psychiatric unit and attempting to force him into unsuitable mainstream secondary. NHS will be reluctant to get involved because they would be asked to chip into the tripartite funding.
To get a residential educational placement you have to prove that she will need 24 hour care for social as well as educational reasons. So, for Aspergers, you could state that she needs to be proactively taught indepence skills in a safe 24 hour specialist setting.
Report collecting is the key. I asked psych unit for a report stating all DS's difficulties, how tghey affcet him and what he needs to overcome them. PU would not state explicitly he needed 24 hour residential (as they would be asked to contribute) but framed the report in such a way that it was an obviosu need. I also obtained a private educational psychologist report that did explicitly state need for 24 hour assessment.
You'll need to do most of the legwork here in finding a suitable indpendent residential school, asking the LA for that to be named on the Statement and providing the proof that such provision is required - as the LA will obviously try to downplay her difficulties to save money (as they already seem to be doing).
If the LA / SS / NHS drag this out too long you'll need to think about an early appeal to SENDIST so the hearing is held prior to her 16th birthday if possible.
So, collect reports, find a school, ask for Statement to be amended - appeal if necessary.