OP you might find it easiest to decide on your preferred area and then look...otherwise you will potentially have too many options to consider.
So I would suggest determining which area works best for commuting/having access to all of London's amenities but equally being able to access parks and out of London attractions quickly too. Will you be getting a car or getting around on public transport?
We live in SW London and love it. In Wandsworth/Merton there's good access into London via Northern Line, District Line, overland trains (to Waterloo and London Bridge and via Thameslink to St Pancras), trams to Croydon, buses to Richmond, Kingston, Epsom, Sutton, Brixton, Croydon and into London (Vauxhall, Waterloo and Victoria).
Lots of good schools locally and in surrounding boroughs (Coombe Girls, Glenthorne, Graveney, Greenshaw, Ricards Lodge, Burntwood, Lady Margaret's). Some schools are very strong on music/arts (Glenthorne and RL specifically). All do languages and for most Spanish is an option.
Have you visited London before? Do you have any gut feelings of what you'd like in terms of neighbourhood? Some areas are more family oriented than others. Some boroughs (and schools) are more spread out and transport links may not be as good.
In London it's quite normal for DC aged 11+ to travel anything up to an hour to go to secondary school (even within their own boroughs). Not many have the luxury of going to a school that's on their doorstep but it's possible. You may just strike gold.
I think within reason I'd almost be tempted to determine the 'quarter' (NE, NW, SW, SE) of London you'd like to settle in, and then do research based on what you can afford to rent and/or best school options. That will focus your thoughts.
Any possibility that you can move before the summer hols so you can look at some of the schools/areas before applying?