My daughter, who lives in Austin, really likes the Apex London Wall, which is in the City of London (the original London). It is on a small quiet side street, is modern and very comfortable.
Moorgate tube station, on the Northern line (as well as the Metropolitan, Circle and Hammersmith and City) is near by and so is Liverpool Street Railway station (central line).
It is correct that the Norther line Bank branch will not be running south of Moorgate for some time, but most of the places you would want to go south bound are walkable and it is working fine north bound.
It would also be a shortish walk to the Tower of London, Tower Bridge and the Thames.
There are lots of restaurants nearby from very grand to ethnic fast food. There is also a charming one (or possibly two) off coffee shop less than a 5 minute walk.
The City is a bit off the tourist radar but is definitely a place where people live and work and it would not be a problem coming home late. In that sense it would be more relaxing than Camden or Kentish Town.
It is likely to be slightly over budget (I came up with £201 pn for 2 people in a twin room staying more than 4 nights) but it certainly gives the "I am on holiday and being pampered" feel.
The Apex Temple Court is popular on this forum but would not be as good for getting to your daughter's course.
I am just over your age and have lived in the City for 30 years (when I moved from Cambridge to London I reckoned that if I had to move to London I should live in London proper) and have remained pleased with my choice.
The City is also an interesting place to wander in being full of historic surprises.
Edited: 28 October 2021, 14:18