Not a huge amount to add but I'd recommend using the app City Mapper to help you with getting around. Google Maps works well too but City Mapper can be smarter about timings, alternative routes and helpfully tells you which end of the tube / lizzy line train to be at for your exit. For the latter, the platforms are so long that it really pays to be at the right end.

If just walking around, then Google Maps is better as you can put the phone in your pocket and set it to vibrate when you need to turn.

Google "London hop-on, hop-off bus tours". For a first timer to London worth doing, even if very touristy. Are several operators

No, I did mean City of London. A good friend who travels way more than I do recommended that. But I may take a better look at the attractions we want to see and maybe another neighborhood will be better. It doesn't seem like City of London is going to be any cheaper than other areas, so price probably won't be a reason to stay there.

Its a business district and so it can be cheaper but at weekends not weekdays, randomly selecting the Hilton in march is consistently circa £350/night of a weekday and £175 for a weekend whereas say the Southbank Hilton is more like £250 weekdays and £225 weekends but is much more variable

London is more blended in districts than some other cities and so the rules arent as hard and fast, certainly in recent years the areas like the City and Canary Wharf are vastly more busy at weekends than they used to be, both used to be a ghost town of a Sunday. They arent terrible places to stay but there is a definite premium in the week which I'd question the value of paying for if there isnt something you actively want to do in the area.