I added a message then removed it because a bit insulting to one of the schools. However, I'm back again and feel I must agree with Xenia to extent, if you're after tip top academics and exclusive feel then St Pauls or Westminster are the ones people go for, also City of London, marginally less exclusive but 1st class academics. However would like to defend NHEHS here, it is v academic and a fun place to be with the caveat that it is decidedly down to earth and not at all flashy or glam, like a grammar school but not for free! Like all GDST, girls are from a variety of backgrounds and I think this is a good thing but if you want the full on, big name public school experience then it will not be for you.
Arts Ed is primarily a performing arts school if this is what you want it is v good and I know several children who are v happy there.
Harrodian is v glam and exclusive, not famous for its academic achievement but has nice grounds and facilities. We didn't apply because of the off putting £100 reg fee!
St James has a very unusual spiritual/ethical approach, if it appeals to you then you will be happy there as some friends of mine are. Will not appeal to trad types.
Latymer is my personal least favourite I have heard reports from friends with children there that it can be aggressively competitive and also that because of all the rich and famous pupils and parents you can feel like the church mouse. I went to the open day in 2006 and the head gave a v aggressive speech
'this is a school for clever children, the cleverest children in west london' he boomed to a somewhat stunned hall full of parents, altho in fact as Xenia also pointed out the cleverest tend to go elsewhere.
Referring to former colleagues he assured us that
'we've been weeding out the deadwood'
Still it remains v popular round here with highly competitive entry and v glam/super sophisticated once there. I also have friends who are happy with it and know nice kids who go there.
I notice you haven't mentioned Godolphin and Latymer - a fantastic school along the lines of St Pauls Girls. Hampton boys is also excellent on all counts. Ibstock Place and Emanuel also worth a mention, considered 2nd league academically round here but for that read they actually have taken on the task of educating people who are not geniuses and are in fact doing a very good job - children I know who go there feel valued and much is expected of them. Ditto Francis Holland/More House.
Good luck with your search, West London is a war zone when it comes to 2ndary schooling, your children must be immaculately tutored geniuses and/or you must be filthy rich/extremely poor or prove yourself a Christian, preferably Catholic. If you can't manage this you will have to move/send kids off to board/grit your teeth and send 'em to local non church state.