I'm afraid that B/Ham KE foundation is quite different to appeals in other areas of the country. The criteria for entry is strictly on academic ability as shown in the exam unless there are special circumstances. Exceptional sporting ability will not help.
I'm afraid we get this every year - lots of very bright kids who for whatever reason don't seem to perform to their potential in the 11+ exam.
I have to be honest with you, unless there are particular extreme circumstances for underperformance in the KE 11+ exam in B/Ham there is frankly very little chance of getting through via an appeal. By this I mean something like a severe illness, recent death of a close relative etc etc.
The KE Foundation handles the appeals process quite differently to other LEAs such a Bucks where a significant proportion of appeals are successful. In KE B/Ham is only 3-4 per year that are successful and most of these relate to an appeal to move from one KE school to another
Only 3 Appeals for ALL of the KE Grammars were successful last year:-
Demand for the Schools remained strong and the number of applicants who took the test in November 2009 for entry in September 2010 was 3,781 (up from 3,585 in the previous year). This compares to the highest ever figure of 4,106 applicants for admission in 2001.
The number of admission appeals pursued to a final hearing was 84 compared to 96 in the previous year. Three appeals were allowed (five in the previous year).
You can view the statistics via the Governors Report on the KE Foundation website http://www.kingedwardsfoundation.co.uk/documents/FO369_2010.pdf
You best chance is going to be via the waiting list - given the scores there is a definite chance but don't go for the appeal route unless there are bone-fide reasons for underperformance.
Sorry I can't give you any positive encouragement but it's best you are aware of the facts