As has already been mentioned, you need schools with places. The most popular schools will be unlikely to have places, and most of the schools on this thread will fall into that category. Living close to any of these schools will put you high up the waiting list, but that isn't of any use unless a place becomes available. There is unlikely to be much movement now and you ideally don't want to have to appeal, as that will slow down her starting.
Flip things around and find out where has places. Ideally a school with a few places or a couple of schools nearby with places, so places will likely still be there once you move. You may just need to choose an area and get an address so you are ready to apply. As an aside if you are looking for rental, that is a competitive market, so I would get started on that and not overly focus on a particular school. If a school has a place and no waiting list, they will give you the place regardless of the distance you are living. Someone mentioned Haverstock had places, which might be worth a look. Hopefully there are other schools.
Not sure if you are aware, but Catholic schools are often popular even amongst those with no particular faith. This makes them more oversubscribed. Other areas known for good schools will attract families and also be more oversubscribed. For example the Borough of Sutton (not especially convenient for you) apparently has no spaces in any of it's secondary schools according to it's vacancy checker.
Given schools start in 5 weeks, I would start pretty quickly looking at rentals, find out about spaces nearby and apply as soon as you are able. Hopefully you will get a place you are happy with fairly quickly.