OP, your plan sounds fine to me.
Our DC did not apply for loans. We are both economists and are very aware of the strains on public sector spending. We could afford to pay for our DC and felt we should. The fact they leave University debt free is a bonus, and will make it easier for them to stay in London where they may well need as large a mortgageable salary as possible.
(When this topic comes up there is often a stark contrast between Londoners and others. Presumably memories of being absolutely cash strapped when buying that first family home.)
What you give her will depend on how much is included in her accommodation, and the sort of person she is, and the friends she makes. Is she the sort who will want to hang out in coffee bars with her friends, or will she be happy to bring along her own thermos with friends doing the same? Is clubbing a major social activity or is she happy to look forward to a big night out every few weeks and in the meantime hang out with friends in each others flats or go to 'Spoons.
Both DC were part of the latter group. DS belonged to University societies, played some casual sport, and hung out with friends. He also does not drink which saves a lot, and studied quite hard. He was content with £1,000 a term with some top up from grandparents and some on-campus earning. (Do musicians have the opportunity to earn from casual gigs?) Ethnic food can be cheap in London, and DS and his friends regularly descended on China town. I am also a bit confused by the comment about transport in London being expensive. DS tended to walk. DD, who is in London this year, would have cycled had she been able to go into University.
DD gets £1,200 a term in Bristol, which is fine even though she runs a car. There are some with an awful lot more, but her friends tend to be sporty or outdoorsy and on about the same budget. The only time she had a problem was the first term when she had a lot of initial sports purchases (she had to buy a blazer as well as a full kit, plus gym membership and hefty subs) but this was solved by giving her a top up for her last week.
Observation suggests that whilst some students are good at spotting bargains: charity shop crockery, buying rail tickets in advance or using the coach, knowing the coffee shop that charges £1 as long as you bring your own mug, returning the router when you leave your flat and using USwitch to decide utilities, etc, etc others expect to live the same life as high earning parents.
My advice would be to discuss it with your daughter and start lowish (£100pw perhaps) and suggest you will pay for big ticket items or raise it if it becomes clear that she has regular additional expenses, perhaps music related. I don't think either DC feels they have missed out by having to budget, indeed I think they see it as a useful life skill. And as they get closer to graduating (both will end up as doctors, one medical, one academic) they are please that there will be a clean break with no loans to think about. when making career choices.
I doubt many will agree, but feel it useful to offer the alternative view.