So sorry your DS was so ill. It must have been very distressing.
Is this school the closest to your house?
To win an appeal you have to demonstrate that some specific qualities of this school are what your son needs due to his social/ medical needs. Or his interests etc.
So, for example, if he had any form of counselling or psychological support for the distress and loss of confidence, you could ask that person to write to say what you have said about confidence/ friends / support.
If he gets very tired as a result of his treatment or condition, you would need a doctor to explain this in a letter and say that it is important that he does not get tired out by travelling and as the easiest journey this is the best school for him, Maybe there are other facilities at the school that make it suitable in the light of his needs: an experienced pastoral care team that look after children like your son, quiet places to rest, etc. Maybe the school has an indoor regular club for children who like / need such a thing, socially or physically.
Letters from professionals need to be specific , to name the school and say why that school is the best school for the child, with a reason linked to the medical / social need. They need to offer this opinion as their own, not say 'xxx's parents feel that this school is best' or 'xxx's parents tell me that,,,'.
For other grounds for appeal you again need to show that specific aspects of this school suit your child and he would lose out by bit having access to these facilities. Another thread gave the example of a child who has an aptitude and passion for maths having grounds to appeal for a school that has maths clubs, enters competitions and does Further Maths at GCSE where the offered school does not. This can also apply to hobbies, or a particular language that is taught, music, etc.
It might help to explain that in the midst of all that went on you didn't realise that it would have been good to include your medical reasons when you applied.
Do you live close enough to the school to possibly get a waiting list place?
You can go on the waiting list and appeal.