Hi Cosmiclyfe,
I went to King's College as a girl for Sixth Form (I'm in my third year of a Master's degree now so left just a few years ago) and thought you may like some info from someone who's been through it. I vaguely considered Westminster when applying for Sixth Form - I also got an offer from Tiffin Girl's and looked round St. Paul's but decided they weren't for me. I think its likely that the Girl's results will be higher (on average) for both schools purely because it is far easier to design highly academically selective entrance exams for 16 year olds than for 11 or 13 year olds - it was certainly the case in my year that the girls were considered brighter (again, only on average). I really don't think the exact A level or IB results should factor too much into your decision though. They are both fantastic schools and its far more important that the school environment seems suitable for your daughter. I ultimately chose King's because it was closer to me, and I'd also heard a lot of rumours about Westminster and their 'party' scene. I also think that prestige is not necessarily a good thing, as it comes with a sense of entitlement that is not necessarily justified - this was slightly evident at King's and is definitely rife in other top private schools as well. What I can say for King's is that is has a huge range of extra-curricular activities and the drama department has a massive annual budget - the shows I saw there were fantastic and way above the standard you would expect from a school. Another great thing they do is run a program on Friday afternoons where you do some form of volunteering or other activity (CCF is an option). I was a teaching assistant in a Special Need's school for a year and it was an incredibly opportunity - not just for my own happiness but its something that will always look fantastic on a CV.
The entrance exams and interview at King's are not particularly fun, but there is nothing unseen on any of the papers (except the General paper which can't be prepared for and just requires a bit of thinking) and the interviews in your chosen subjects are fine as long as you can show you are passionate about your choices.
Any sixth form transition will be quite tough, as the step up from GCSE to A level is perhaps bigger than it should be, and I can also say for certain that parties will happen anywhere where teenage boys and girls are suddenly in the same place. The main thing that put me off Westminster was that, at the time I was applying, their parties were well-known for being considerably wilder.
The teaching at King's is portrayed quite accurately in the guides (although I think there have been quite a few changes in staff since I left). For a very academic school it is surprisingly relaxed but still well-structured, although the IB caused students a lot more stress than A-levels did.
I am by no means saying absolutely decide on King's - I have both good and bad memories from my time there - but I am saying that despite the times when things weren't so great, I don't regret my decision at all. The new girls are really well looked after when you arrive and there are loads of school-organised socials before the term starts to make the whole process less daunting. You also get put in an all-girl tutor group as well as a mixed form group, so you don't feel so outnumbered.
I'm sorry I can't offer much information on Westminster, but thought a bit of insight into King's may help. I also should apologise for the length of this post - I was procrastinating some tedious lab report when I found this thread :D
If you want any more information I'm happy to help the best I can :) And good luck to your daughter I'm sure she'll do great whatever she decides