My niece studied at uni outside of London, then came back to London and got her first job.
she rented in Paddington to start with, loved all what was on offer but wanted her own place, Paddington was way out of her price range.
She had her heart set on a flat in the Kings Road, but service charges and a strongly worded discussion with her solicitor put her off that thankfully.
She has a moped so transport wasn’t a problem to her, and she started looking “out of the box”, and found a tiny Victorian two up two down in Peckham in a beautiful street, and something really, really drew her to it.
it was well within her budget, it felt extremely safe, the neighbours are lovely, it’s 3.5miles to Tower Bridge, and London Bridge, there are buses in abundance too.
she has been there a couple of years now, she is very very happy there, she has the second bedroom which she had a lodger in for about a year, before they bought three doors down.
The only problem with it, is parking a car, there is no parking whatsoever, if you have a moped you are fine as you can park in your own small front garden/hallway (a runner from the front door to the hall, and a piece of wood for the stand), and she does keep her bike spotlessly clean, she is very proud of it.
A lot of people say my niece is living the London dream, and if you are happy with what your daughter wants to buy, (would you buy it?) would she be able to live the London dream, or would it be better to find a two bed so a lodger can help with bills.
I wish you and your daughter all the best.