There isn’t a lot on rightmove for students atm (wrong time of year I guess) so it’s hard to gauge.
There is not really a student market as such in London. A University/SU might have some registered landlords, but mainly it is a case of going into letting agents to ask if they have flats where the owner is willing to take students. In a tough property market it is a rare landlord who won't. At one point students formed the largest group for Mayfair rentals. (Overseas students, even if they are not out and out rich, will prioritise safety and proximity, even if it means sharing a room. They are paying a fortune so are here to study and want to be close to the library.) Though you might have to pay six months in advance.
Plus it is absolutely the "right time of the year". There is no committing months in advance, usually no more than a month in advance. Students hit town the same time new graduates/first jobbers, so the market is always busy in the run up to September. A sensible student would start hunting early, say June, and then get a job to pay rent over the summer.
KCL is good in that SE London is relatively cheap, though can be gritty. I rent out a nice 4 bed house in Greenwich, by a station and in a nice residential street, and get sharers who pay £550 pcm. I really do not understand the prices quoted here. There will be cheaper properties, above shops etc, further in.
I am also realising that people in London live differently. Unless you earn a huge amount you are likely to be property rich and cash poor. I have, for various reasons, spent the summer out of London realise that a lot of people have a lot of disposable income. Cars...first no one seems to walk anywhere, whereas I am used to walking 40 or 50 minutes rather than engage with traffic or the Underground. All my London neighbours drive old bangers, if they have a car at all, even though some have impressive careers. I find it strange that so many people drive new cars, especially white ones which need cleaning every week, and don't get me started on personalised number plates. There are lots of other things. Bumping into a billionaire I knew from the school run, in a London Lidl looking just like anyone else, eating out in interesting ethnic restaurants, rather than somewhere "posh", not knowing anyone who belongs to a golf club. Obviously there are lots of exceptions, but frugality is normal, and displays of wealth are a bit off.
It might help students thinking about London to take a step back. What do they really want. Replicating a "typical" student experience can be expensive, but there are other experiences. Lots of students will be focussed on their subject, and there is a wealth of opportunity in terms of public lectures, etc. (DS was part of a group doing a voluntary piece of research in his first year, and one of his professors opened their address book and gave them introductions to an impressive array of senior banking and government people. DS did not really appreciate it, whilst we were gobsmacked.) It will probably be dim sum for a birthday celebration rather than a pizza in a chain restaurant, and a student disco or society, rather than clubbing. Trawl charity shops in posh areas for bargains. Got to student plays and art shows. There is plenty that is free or cheap, but a frugal mindset is key. As with adults, you need to be seriously rich if you want to replicate what people outside London may consider "normal".
Frist year in halls, and find out where second and thirds years are living. With luck you might be able to get a hand-me-down flat. DS took a room in an existing flatshare that had been going so long that they still had a Betamax recorder on the inventory.
And don't expect an ensuite.