Hi, are you still looking for recommendations?
I did a huge search over 2 years of my daughters (we live abroad) to find a girls school that was like you say. I would highly recommend Benenden School, in Kent like heronsfly suggested. If you want a traditional British girls boarding school it is the only one I found that fitted this description.
Reasons: it is the only all-girls school I found that is 100% boarding - they could keep this because they have such a high waiting list (other schools have had to take on day girls or go co-ed in 6th form or take in significant numbers of overseas students). They do not advertise as they are known by word of mouth.
Other thing I looked for since I am sending my girls all the way to the UK I didn't want an international school (otherwise I could keep them here) so Rodean has a very high percentage of girls from Hong Kong and Cheltenham from Arabic countries (we have roots in both those places so I wanted a more local UK atmosphere). In Benenden they limit overseas students to 20%: 10% UK citizens who are expats, and 10% foreign students. For me this was the reason I sent my daughters to UK, otherwise, like I said, they could go to an international school, like they did for prep school. Very importantly too: as it is 100% boarding, the girls have very well organized weekends which when I look round no other girls school has as varied and interesting. It is sad when half the school leaves on the weekend for the girls whose parent are far way.
BUT be warned - places are so over subscribed that you must put your daughter's name down as soon as possible unless she is entering for a scholarship. In the past it is known that there have been 4 applications to one place; Another thing - although it is difficult to get in because they can choose the best of many candidates but once a girl is in it is not at all pushy on the academic side at all. It is more encouraging in everything. The music is brilliant, theatre too, art and sports good too. It is also rare to find a girls school which is such brilliant facilities - like boys public schools. Their science centre is one of the best in the country.
I urge you to visit it. The whole of Benenden School is on a campus with 200 acres of land in Kent (which is something fantastic). There is a lake, eco-classroom. The sixth form buildings are great. Many tennis courts, indoor swimming pool, gym, squash courts, etc. In latest Ofsted Report it scored excellent in every single category.