Good to hear you're planning a taxi from King's Cross to Waterloo hotel, they are *not nearby each other, and Sundays are rough days to find transportation across the Thames, the last time that I went. The Treasure Room in the British Library is right near King's Cross, you may want to do that first, if they're open on Sunday and at the time you'd be arriving. Double check everything you have planned on Sunday, I found not much was open. Calling that day 1 is kind of generous. Won't be a whole, useful day. You'll have great night views staying by the river, tho. :)
I bring a celestron monocular (one half of a binoculars) when I travel to London, not only for sightseeing, but being able to read details, tops of huge artworks, and departure boards in train stations so I'm not running to see what I'm missing. $15 on amazon, nifty for a traveling astronomer like myself.
HoHo buses saved my last trip when there was a protest rally shutting down regular transportation. Their staff worked to help all other companies' ticket holders, friendly. When I lived in London for 2 months, I used the city buses often, and this was nicer to have specific sights as the stops, instead of having to count/track all stops. I have pain/limitations, so sitting downstairs, avoiding crowds, getting some quiet and having no concern about pickpockets were positives for me. Everyone else will probably say bad things about them, but they can be useful (and lower stress) to some of us.
Victoria & Albert Museum could probably take days :) in a good way. Their jewelry room was dazzling, glad I saw it *after I saw the crown jewels, or I might have been disappointed in them, after seeing the V&A. Harrod's was a thing, just didn't care for the materialistic energy of the people there, and I must not have had the best map of the store, couldn't find much of what I wanted to see. Others must have better luck than I did.
Westminster Abbey, & Churchill War Rooms are nearby each other, but Borough Market is NOT nearby. Use Google Maps often to check distances, and plan all in one area at a time. Parliament is opening back up again for tours, that would be right next door to Westminster Abbey, could be good. Also Trafalgar Square and its galleries are close, too, also they're about a 6 block walk from some of the theatres, so it might be something to take a peek in on. Free is good. I found some family history people in that National Gallery, it was a very different experience because of that.
As you're hopping from one side of the Thames to the other a lot of this time, you may want to look into the boats that work off of Oyster cards, if weather allows. If it were my trip, 15 days in Scotland would be too many, with only 3 1/2 days in London. On that 5th day are you going to Heathrow early, by big taxi or train ride? I would move at least several days from Scotland to London, if you really want to see most of the things on this list. The way first -timers drastically underestimate London is unfortunately consistent. I also prefer to stay the last night in Windsor, see the castle, and only have a quick hop over to Heathrow that next day, like 15-20 minutes to get near the airport (tho it is its own city in itself, of course).
Keep researching, mapping, and don't underestimate London :) I'm a NYer and could see the sights in a day or two, but London's still a challenge after living there for 2 months... I think I need about 6 months there some day. :)
Have fun, there are good YouTube channels that show the walks and sights too, lots of resources and virtual tours are also online now.
Happy travels!
-Alison