Wondering what would be a good hotel for the stay under £120?
We would prefer chain hotels where basic things like good sumptuous breakfast (must have)
Chain hotels costing under £120 and sumptuous breakfast do not go hand in hand. In fact even most of the chain hotels costing more don't provide sumptuous breakfasts, at least not what I would regard as sumptuous. Premier Inn breakfast is mediocre, Travelodge is awful, Marriott, Hilton, Accor etc offer varying degrees of quality depending on the particular brand of each company.
The best breakfasts are to be found outside hotels in London but of course this could eat into your sightseeing time unless you get up early, have already identified somewhere for breakfast and is close to your first sight of the day. Ok, it's less convenient than rolling up at the hotel dining room but if by sumptuous you mean a genuinely good breakfast then you'll not find it in a hotel in your price range.
Also, with four nights you'll struggle to scratch the surface of London, when you say "cover all the major tourist attractions" what precisely do you mean. People have been travelling to London for years and still not covered it thoroughly. I have the luxury of being able to get to Londoon Waterloo within 1hr 15mins which does permit me the luxury of taking things easy when visiting as I know I can be back at any time however, take the Imperial War Museum for example. Not a far walk from Waterloo but several hours spent exploring the museum, something to eat and then a walk back to the station and I'm looking at mid afternoon and that's just one sight.
London is absolutely crammed full of major sights, then there's the not so major, then the specialist sights and everything after that. I would say that 4 nights is barely enough to cover just a handful of the major sights in any meaningful sense.