Welcome, JaLou - and I'm so sorry to hear what you're going through.
I wish there was an easy answer - but there isn't!
Appeals can be very stressful, and realistically most of them aren't going to succeed - not least because all sorts of factors might affect the outcome, some of which could be completely outside your control.http://www.elevenplusexams.co.uk/appeal ... school#c20
You will probably want to go ahead with an appeal if only to reassure yourself that you have done everything possible. I would suggest, though, that the less your daughter knows about what is going on, the better! Best to be very vague - you could tell her that you'll submit an appeal, but that it might take until July before you hear anything definite.
You will need as much academic evidence as possible, and don't overlook reasons for wanting a place at the school being appealed for.http://www.elevenplusexams.co.uk/appeal ... cation#b11http://www.elevenplusexams.co.uk/appeal ... -school#c2
The panel will obviously look long and hard at the evidence for maths. Jumping two whole levels between years 3 and 5 is very encouraging. Late evidence can usually be submitted up until a fortnight or so before an appeal, so she has time between now and the summer term to improve even further.
I think it's a plus point that the headteacher is willing to attend the appeal with you.