MN164's list above is pretty exhaustive and should provide you with an excellent starting point. As she says, it's wise to look at a range of schools. Most heads will recommend that you have at least one school that's 'aim high', one thats a likely thing and one 'fall back'. The exams are very competitive and results can be unpredictable. My advice would be to go and see some/all of these schools by yourself (not with your daughter) when she is in Year 5 then make a list of the ones you like best, talk to your daughter's current head about where she realistically might be suited to, refine your list again and then take her to visit the shortlist schools when she is in the autumn term of Year Six and see what she thinks.
My daughters went to primary school in Hampstead and most of the Islingtonians went for City. It's an excellent school and I would say that latterly it's been considered a notch above South Hampstead (but as Lynharvey says above, that may all change now that the building works at SHHS are finished). City has quite a different vibe to SHHS - more creative and egalitarian, I would say. It's less local in its outlook and girls will come from far and wide to attend. SHHS is more local and perhaps more social. There is quite a big social scene centred around Hampstead and lots of mingling with UCS boys. That's fine, but it does start quite early in Y7.
I would explore the public transport options for all potential schools rather than thinking about distance by car as by secondary school your daughter will probably want to be travelling alone, as per most of her peers. The 271 bus goes from Islington up to Highgate/Channing but bear in mind that traffic on Highgate Hill can be appalling in the morning rush hour. Highgate is incredibly difficult to get into at the moment as it's one of the few co-ed independent options and they don't seem to be able to get their admissions quite right - last year they had to take an extra 'bulge' class as fewer places were turned down than expected, but a couple of years before that they didn't offer enough and had to call back kids who hadn't originally been offered interviews. Friends whose daughter started there this year says its quite traditional, more so than they expected. Channing is smaller and cosier and seems to have a very happy, well-adjusted vibe.
For SHHS you can get the overground from Highbury and Islington to Finchley Road and Frognal then it's a short walk. Friends of ours in Islington have a daughter who goes to SPGS this way too - she gets a Clapham Junction train to Kensington Olympia then walks down to Brook Green, so although geographically it's quite a distance, it's a pretty simple journey. I know that there is also a coach that goes there from Finchley Road (ditto NLCS) so it might be that you could get your daughter to the coach stop in the morning and she could get to school that way.
One thing to bear in mind with the more far-flung schools is that if your daughter is sporty she is likely play lots of after-school fixtures and this can mean she will need to be picked up from school at a later hour i.e. my older daughter plays a lot of netball and after 'away' games sometimes returns to school at 6.30pm. When she was in Years 7 & 8, I felt this was a bit late for her to be facing a long journey home alone in the dark (during autumn/winter terms), so her dad or I would have to trek off to get her in the car - not always very convenient.