Bluntly, it’s not going to be easy to find a good school place at such short notice.
State: the council will have to find your DD a place somewhere, but not until you have actually moved, and it will be whichever school in the borough has space (aka the least popular school in the borough, usually for good reason).
Private: you will need to find a school that is not full, or has an occasional place coming up, ie someone is leaving. Tbh occasional places at the more popular schools will likely have been taken by now, as the pupils leaving will have had to give notice at Easter and those schools have lists of people waiting for openings so probably went to the list first. But worth phoning around. You could also ask schools that have a 13+ entry (eg Kingston Grammar, Epsom), the exams for 13+ entry have long passed but they might just have had someone who was holding a 13+ place but decided not to come.
Private schools that are not full: I would suggest looking at some of the more recently opened/rebranded private senior schools. For example the new Thomas’s senior school in Richmond, or the London Park Schools.
Sorry I am less familiar with N London schools but same general principles would apply.
There are school entrance consultants who can help you with all this, for a fee. They are mostly focused on private. They will do the ringing round for you and will have a more comprehensive knowledge of what schools are out there and likely to have space.
Be careful, if a private school has lots of space, it may be in trouble financially.
Also, I would factor in more like £30k, with 3-5% increase each year. Sorry 😬
I agree that looking out in the Essex/Suffolk direction may work better if you’re not wedded to London. Good London secondary schools are ridiculously oversubscribed right now.