I'm pretty sure the appeal panel will be totally independent, to make sure the governing body is behaving reasonably. Otherwise there could be governing bodies doing totally daft and discriminatory things - the appeals procedure is there to stop them doing that. The school or LEA really should inform you of the appeals procedure and what to expect.
As I understand it, there is a diplomatic tightrope to be walked: you need to say all the good things about the chosen school, without slagging off the offered school, just pointing out all the reasons the offered school isn't suitable for your dd.
Some other thoughts:
Can you get a letter from your GP - it might carry more weight than one from the school nurse? IME GPs are happy to help with this kind of thing, why shouldn't they - if they don't help you, they're more likely to see more of you!
And if there's anyone else you can get a letter from, then ask them, eg educational psychologist, if you've ever gone down that route.
I'd say also you need to rack your brains and think of every possible reason why your chosen school is better for your circumstances than the offered school - some examples would be (not knowing the details of your situation):
DD, and ds, and you, have established friendship groups at the chosen school; dd, ds, and you, know virtually nobody at the offered school.
Transport factors: you can't get dd to one school and ds to another. Going to the offered school will mean using a car/a difficult bus journey/an expensive train journey/a change of transport. DD could walk/travel independently to chosen school when a bit older.
Breakfast or after school club provision - you may not use them now, but can say you plan to use them later on
Better sports facilities at chosen school - dd keen on sport
A particular club, eg drama club, at chosen school - dd keen on drama, good for her personal development, self esteem
It must be so annoying, having to do all this! I reckon however - from what I've read and from speaking to other parents - that if you are persistent and hang in there, you can get the place you want. Especially if you cling on to a place on the waiting list - I reckon lots of people give up and accept the place they're offered.
Best of luck, thinking of you.