I think you should buy the RS London or England guide book. First trip to the UK or to London, I would stay in London for most of the trip. 9 nights seems like a lot but it is not when there is so much to do and see just in London. If you want to do a day trip out of that time there is plenty to pick from, Windsor Castle or Hampton Court Palace or Bath. But one day trip or two is plenty. You want to see the biggies in London. What you want to see would take a two week trip. What you are listing above, you will not be really seeing London. The RS books will list hotels and that way you can see on their websites what hotels will fit your budget and needs. Just pick a hotel that is near a tube stop and you will be fine.
For research on what will really interest you, watch free YouTube Videos on London and England including RS videos which are on YouTube. Read various guidebooks which you can get for free from your local library, the internet is a great source to look up sites, musems, cities, images of what they look like, facebook, instagram. It will give you an idea as to what places look like and what interests you. What I am suggesting below I have done on two different London trips and can't wait to go back to London again and see musuems and sites I have not seen yet.
Plenty of museums to visit, British museum, Victoria & Albert museum, the National Gallery, Courtauld Gallery, British Library and Churchill War rooms ( not to be missed) and the Wallace Collection, St Martins-In-the-Field Church.
I went on the Royal Day out - ticket called that for Buckingham Palace where I saw the state rooms, the Queen's Gallery ( paintings) and the Royal Mews- which are where the carriages are. It took the whole day! You can buy other tickets to just see the state rooms or the state rooms and the gardens.
I also toured Kensington Palace and that was wonderful.
Walking tours of London, I went for the first time on the Original London walks and it was great. Went on pub crawl and another walking tour. I also went to the theatre one evening to see Wicked.
Harrods, Forum and Mason, Selfridges are all great to see for window shopping and shopping.
Read up on how to use the tube, very easy, I also did screen shots for the tube stations from my hotel to where I was going every morning as I knew where I would start the day out with what museum or site.
I hope this helps you in your planning and get a better understanding as to why everyone said not to go to Scotland and not to try to do so many day trips. It can be a bit overwhelming, but once you figure out what you really want and can see in a day and map out how you will get there and understand the city better, you will feel more comfortable as to the planning of the time in London.
London is a city where you can go to many times and still enjoy and not be bored.
Have a wonderful safe trip.