Whilst I have no doubt that you have landed upon this forum in an attempt to seek informal advice, you have come to the wrong place. This website is populated predominantly by students, and this one in particular is for those who are looking for advice on legal careers. So the vast majority of members on this site are not qualified to give you legal advice on this, and should not do it, even informally.
As it happens, I am a practising barrister and I deal with cases like this (breach of contract cases against independent schools) regularly. However, for hopefully obvious reasons I am not going to give legal advice on this forum on a matter like this, where much can turn on individual facts and terms, but where in any event I am not officially instructed. One thing I will say is that this is a highly specialist area. There are plenty of lawyers who can give you advice on breach of contract claims. There are few who specialise in education cases particularly, so even seeking informal advice from lawyers on the internet is something you should be wary of (and those sorts of forums do exist). In reality you need to seek specialist advice on this now, and should not be looking for it for free on the internet.