If you're not too fussed about which bit of London you live in, and can afford central, then I'd be looking to move to the Battersea/Clapham area, because them you get good connections to
Westminster and KCS Wimbledon (you need to check about transfer to senior department, and when you need to register by, plus exam timetable). Single sex, but girls in the sixth form.
Dulwich College would certainly be worth a look, if inky for a point of comparison, Alleyns is the co-ed one of the Dulwich foundation schools. They're competitive 11+ entry, you just need to register by the deadline. BTW these schools run a joint coach service, so they are easy to reach from whole swathes of south and central London
Emanuel is changing its entry procedures from Sept 19, and all those who enter within the application should window (will be some time in the autumn of the year before entry) can sit the exam, but they no longer guarantee to interview all applicants. They have a new system of capping attendance at open mornings at 2000, so book!
Ibstock isn't that well served by public transport, so only consider it if you want to live somewhere handy for it. Harrodian slightky better served.
Portland Place has a fairly mixed reputation, but is very central (stone's throw from Oxford Circus)