In a sense you have answered your own question. If the IB is your objective and the reason for the Appeal then you should focus on exactly why that is the best education for your daughter, and ask the Head to do the same in his/her letter. I would tend to include academic evidence, even though she has qualified, because it may show why the IB is particularly relevant.
I am not that familiar with the IB, so you will need to do your research. I am sure that a Search on this Forum will pick up some useful points to start you off.
If you can demonstrate that there are strong reasons why it is important - in family terms or for other reasons - then it makes this a very interesting appeal case. I think it is a first for us, and it would be one for us to add to the Appeal Q&As if you win, so please don't forget us!
As a note of interest to appeals in general for Kent, I noted in the local paper that a Kent cllr was quoted as saying that there was not a shortage of places at Grammar schools in the County but that children living near to a 'mixed ability' school were given the mixed ability school to prevent children from travelling distance.
Kent Cllr. needs to read his own Council's website! Discrimination of this sort is illegal under the School Admissions Code.
Every school has specified Admissions Criteria and places must be offered accordingly. No one can suddenly say: "Poor lambs - a 40 minute journey is too much, so let's send them to their local school"!
I also beg to differ about there "not being a shortage of grammar school places in the county". Every Kent GS I have ever looked at received more applications than they had places, even since the change was made to results being known before the CAF had to be submitted.