I volunteered at the Girl Guides/Girl Scouts' world centre hostel Pax Lodge for 2 months with my daughter when she was 23 yrs old. She and the other volunteers frequented Camden for both the shopping (fashion) and for the pubs. Easiest educational place (that we recommended to guest staying with us) was the British Library's Treasure Room, a zillion REALLY amazing things all in one big room with comfy benches all around for any tired tourists. The pub across the street was called the Euston Flyer and it was really great, in 2011 and 2015 when we returned to celebrate our Bachelors degrees (hers 2 yrs after mine).
Harrod's milkshakes were also very popular among the staff.
I believe Graham Norton's TV show free tickets should be available for Sept soon, too, unless something changed. That could be really cool for a 20 yr old, depending on who the guests are going to be. There's always a young musical guest too. Very big stars sometimes. If you haven't seen his show, there's a lot on You Tube.
Heather and I did go on the London Eye, we went at night (I had an eye operation when I was a kid and have some double vision and lots of fears of heights, but wanted to say I did it, held onto the bench and went at night to minimize the ability to see straight down). I frankly felt like it was only taking me farther away from the sights, I wasn't impressed, but maybe I missed something. :)
Also, HUGE help, was that I had bought a monocular (half of a binocular, like a short telescope) for travel in 2015, and the details we would have missed, both inside and outside, would have been a long list. It also helped when I had to see the train schedule board when it was WAY at the other end of Waterloo Station, helped to not have to go run for the info. Helped to see the art and architecture all over the city. Amazing difference.
I also have a brother in law and 2 nieces over there in Essex area north of London, and the older niece loves pink gin, which is supposed to start being sold here in the USA soon, but it's SUPER popular over there, strawberry and raspberry flavors, some are sparkling too. When my daughter and I went, we drank hard apple cider, Strongbow. The ales over there are really strong, and if your daughter doesn't love them, there ARE additional, local options that taste better :) And the trivia nights at the pub were also popular with the other 20-something volunteers at Pax Lodge, and the crepe vending cart up the hill, we were in Hampstead.
Gift shops my daughter loved: Globe Theatre, Tate Modern, Buckingham Palace. British Library's gift shop was OK, but not great. NEW gift shop opened at King's Cross for Harry Potter items, looked good online.
I went to Westminster Abbey for a (free) church service, it was Sept 11th's 10th anniversary & I'm a NYer. They just ask that you stay through the whole service. So finding a church to go see will be an easy one :)
Great to hear you have ideas on where to stay that are centrally located. Always good to minimize travel time to the sights you want to see. There can also be many open nights at local universities, if she's considering grad school or anything. Could be fun to see them on the inside.
Have a great time!