I'm looking for a good sixth form with boarding...

If you read any of my previous posts on this topic you will see I am a bug advocate of Oakham. No connection. Just a very satisfied Old Oakhamian. Everyone leaves very well rounded. Happy to answer any specific questions. The Tatler and Good Schools Guide summaries of Oakham sum it up perfectly in my mind.

How about Peter Symonds in Winchester?
State, but has much better results than many private schools and a much respected music department.

Being oversea mum, I think you also need to consider transport link to the airport. Is your DD doing MYP now? Do you want her to do IB or A level? If so, what subjects she is planning to take? Most likely the school will get her to sit entrance exam for those subjects before giving you an offer if you don't have GCSE. All school you listed here a good school, so it is really up to you and your DD preference and also location of your relatives.
So the best way is to write/phone all registrars and check if they have place for your DD for sixth form to start with, if they have place you can start the ball rolling. Short list 4 to 5 schools to visit. I would put Sevenoaks in the mix if they have place for her, if she is doing IB. Both Shrewsbury and Charterhouse are boy schools with girls in sixth form, if she is ok with that then all new girls will start at the same time, may be easier for her to settle in.

Meant to say I believe Charterhouse offers the Pre U to it's most able students, A level and IB this is becoming increasingly common.

If I was you I'd apply the 1 1/2 rule to the family member she's mostly likely to go to at weekends exeats etc. I would also choose a school with a high % of full boarders she will have more fun and that's what coming to the UK to board is partly about. Charterhouse is basically weekly boarding, I know nothing about Caterham, Uppingham Marlborough and I think Rugby are basically full boarding or have a small number of day children, the others I think have a mixture of day, weekly and full. Does she have a particular sport/hobby/musical instrument that she really enjoys? Do all the schools offer it? Does she want to be in a town or rural?
All the schools have good solid reputations, but everyone on here will have their favourite and also someone will be able to wheel out a horror story. In my now extensive experience bar a few exceptions there's surprisingly little to choose between them in tears of facilities, teaching standards etc. What matters is that you and your DD feel comfortable and happy there especially if your not in the UK. My DS is at boarding school, it's not everyone's cup of tea by any stretch of the imagination, but we are happy there, it works for us and that's all that matters.

Do you want proper full boarding (hard to find) i.e. the option of only coming out on Sunday or weekly coming out routinely on Saturday going back Sunday PM or even Mon PM?
Unless you list driving as your main hobby/passion definitely no further than 1 1/2 hours one way less if you can is better and I personally wouldn't put the M25 between me and a school even if theoretically it's only 1 1/2 hours.
You state that your daughter is quite academic competition for 6th form places is quite fierce with many very bright candidates applying from abroad the requirement stated in websites is likely to be the very minimum.

Others with experience will be here soon (we just started pre prep). I'd contact them all, ask for brochures, admittance criteria and how to take peep. Wiki and Google them too. Visit the Independent Schools Association site too, they have inspection reports.

Bishops Stortford College
Haileybury

Good luck OP