Getting back to OP, Imperial suits some students well, and others less well. I went to quite a gentle girls boarding school and at a school event, people expressed surprise that I had gone to the LSE. For as long as anyone could remember the school had advised against London. (It was very much the wrong school for me and I could not wait to escape to the big smoke, something my teachers recognised.)
Catchments for London sixth forms can be wide, and many are used to travelling for school as well as social life, sport etc. I assume it is the same for others who go to schools with wide catchments elsewhere in the country.
Tizer's girls went to a very posh girls boarding school in an era where things would have been more sheltered. If your social life and education happen at school where you also live, it is not unreasonable to want similar at University. Nine stops on the tube or a 20- minute cycle probably feel like a marathon, and you might be happier somewhere more compact.
Our experience of London Universities is that because students are dispersed, the SU comes into its own as a social focal point. Also London is not really geographically arranged but overlayered, so South Ken is student London, tourist London (the museums), French London (the Lycee), workplace London, Prepschool London, Trustafarian London and so on. You can be one and barely aware of the others. Student bars are used by students. Or favoured bars. (DD did most of her very limited Imperial drinking in Hammersmith, a revelation to her as she went to school in Hammersmith, and though she knew the streets, was completely unaware of the parallel student universe.)
In terms of international students, Imperial is somewhere where even the diverse are diverse. DS' friendship group included Chinese from mid-Wales (keen to get somewhere where he did not look different) and north London, educated at a British boarding school, parents who were expats in third countries so educated at international schools, from SE Asia, HK or mainland China, fabulously rich or relying on "family scholarships" (where the extended family support a bright kid to study abroad), mixed race and so on. And stereotypes are changing as more of the younger generation have grown up with affluence and exposure to international trends like KPop. The unofficial social secretary of DS' course was from HK, but they all joined in. Even the British students are diverse, though the north is under represented. Of DS' British friends that I am most aware of, four were Londoners, of whom only one was white, and one was from elsewhere. Two at least lived at home. DD says none of staff at Imperial teaching her were British, though one was Irish.
In short if OPs DD loves visiting London and does not find it underwhelming, she will probably enjoy Imperial. (With the caveat that though there is scope to play hard, you absolutely need to work hard.) Bus pass....will travel. Student London can be cheap, check out TodayTix, TooGoodToGo etc, and there is an infinite variety of things to do.
If you make the effort, join societies and put yourself where you are likely to meet people with similar interests, enjoy your subject and want to put real effort into it, you should be fine. If you sit in North Acton and expect the world to come to your hall, perhaps not.