I have no idea why Rick doesn’t mention this, other than he has his own favorite areas, near Victoria Station, South Kensington, etc. with more of a “traditional” feel in the architecture.
But the City of London ( near the river) and Southwark areas are our favorite place stay.
Right near the Tower Hill Tube station ( not Aldgate) you will find a CitizenM ( just across the road), and nearby a Four Seasonsa Novotel, and a Leonardo Royal Hotel ( used to be a Jury’s). But maybe you would like an Aparthotel—-a hotel, with reception desk and other hotel amenities, and serviced apartments instead of just hotel bedrooms.
The Marlin group has a nice one near Aldgate East station:
https://www.marlin.com/locations/serviced-apartments-tower-bridge