There are 32 boroughs in London and about 200 districts.
That's really hard to pin-point, but most of the theatres in central London are within walking distance of Leicester Square. See the link below.
Soho and Charing Cross are the main areas.
No. The Lake District is about 300 miles north-west of London.
Safe District : Electoral district in which the candidate from the dominant party usually wins by 55 percent or more.