There is a pan-London admissions process so that if you are on the border of a borough, it doesn't advantage or disadvantage you for schools in either borough. Each council can tell you the max distance for the admission on the distance criterion last year (and in past years) - most have brochures you can download with this information. My impression of areas as central as those you have listed is that state school provision is very minimal; it's almost all either faith-based or private, and of course your religion and ability to document it, and finances will determine whether you should consider these.
I'd suggest that if you don't want to pay for schooling and are not going to go for the church route, you might want to consider Islington, Camden, Chiswick etc - just slightly less central. Get the Ofsteds of all the schools near where your decent commuting routes will be, and then get the brochures from the council with distances. But also note that while Ofsted reports are the easiest way to get a snapshot at a distance, they are only a snapshot and are far from the whole story about a school! On top of that, of course they could never capture whether your child will like their teacher, whether your child will like the activities, whether the after-school provision is suitable for you and all that. I was interested in KS too -- it looks like a fairly traditional school, longer hours, emphasis on academics, between the lines more likely to have spelling tests and rows of desks than creative dance ... just guessing though as I have no experience of the school. Also don't bank on one school. If you can choose a neighbourhood with a range of "good" schools in addition to the "outstanding" ones you may be hoping for, this will be a great fallback (as it has been for us).
We did all this, also from afar, a couple of years ago and it was a massive effort; we only partially succeeded (didn't get the ofsted outstanding school we wanted after all that, but got a 'good' one that seems to be working pretty well so far). We had some financial constraints too, and commute constraints.. and it all added up to not really thinking about secondaries at all beyond the vague notion that there are some good secondaries vaguely around here and colleagues in the area with older DC are not terribly worried about it. Our DC are young, and schools can change a lot in 5 years, and what with all the other constraints, it just wasn't possible. Good luck with it! It's hard.