Northbridge House, Queen's College and Aldenham are moderately selective which sounds right for your daughter, especially since you need a space outside normal intake.
These are three quite different schools so you should visit and get a feel. (I don't know much about Aldenham except that it is probably a bit less academic than the other two, which we quite liked, a couple of years ago).
Perhaps add Mill Hill/Belmont? A school I don't know much about, but have heard it has lovely grounds (compared to central London schools) and is quite sporty. Open day on 8 October. It also has an intake at 13, in case you have to wait for year 9 (or your interim solution is not very good).
Applying in-year like this, you will have to be quite lucky for there to be a space at all, so you need to cast your net wide - call the registrars and ask if there is any chance. But spending time on very selective schools like NLCS and Highgate would waste time, and distract and frustrate you - they will require excellent maths skills, no question.
For a (very) rough indicator of how academic and selective various schools are, and to find ideas for other schools, look at pct A/A star at GCSE - e.g. Telegraph online. (And avoid the top 20-50). Good luck!