One thing I found helpful was to try to see it from the point of the view of the panel. They obviously cannot let in everybody who is appealing because they simply want to get into the best school, because it wouldn't be the best school if they did that. When we appealed there were 50 families appealing to get their child into Yr 7 of that particular school: obviously there was no way they were all going to get in.
At the same time, they are usually happy to help special cases and admit a child who really does need this school in a different way from the other children whom they still have to reject.
So your appeal should be based on whatever it is that makes your dd a special case and whatever it is about this particular school that means it is the only place that can meet her needs.
In our case, it was disability, for a friend's dd it was the need for specialist counselling. But it could also be something like a special talent or a special difficulty that can only be met through the special resources of that school.
Be prepared that any claims that you make have to be substantiated. So if you claim your dd has social difficulties and needs to be with people she knows you would need to submit some evidence (from a therapist, doctor etc) that her difficulties are more severe than would be considered normal for a child her age. We sent letters from dd's GP, paediatrician and counsellor.
When we finally got to the private part of the appeal, I was asked to give a verbal explanation of dd's needs and why her appeal was so essential. Be warned: I found it very harrowing, having to explain in front of a bunch of strangers why my dd is different from other children. You may also have to listen to a presentation from the prospective school about how they really do not want not take another child. Again, I found this quite hard, though I understand why it is necessary.