As others have said, if you withdraw your daughter from her current school the LA must come up with a place for her somewhere, even if it means admitting her to a school that is already full. It may take a few weeks, and it may be a school you don't want, but she won't be left without a place.
Another thing to be aware of if you haven't yet given notice is that you are probably committed to paying the independent school's fees for the autumn term already. Most independent schools require one term's notice.
Regarding your appeal, the presence of friends is unlikely to help unless you have evidence from a medical professional or similar saying that your daughter needs to attend a school where she already has friends. The language point is better. That is the kind of thing that can win your appeal. It would help if you can identify some other things this school offers that are not available elsewhere and that are particularly relevant to your daughter.
Unfortunately, the fact you haven't yet pulled your daughter out of the independent school makes it a little harder to win your appeal. Normally, you can win by showing that the appeal school offers things that are missing from the allocated school and that your daughter needs to access. You don't have an allocated school, so you can only really go for things that aren't offered by any other local school. And, if you haven't given notice to the independent school before the appeal hearing, the appeal panel might think that staying at the independent school is an option, especially if you talk about not wanting to give notice until you've got a place for your daughter, in which case the language argument won't help.