I agree about day-trips; there's way too much to see in London to leave the city on a first visit of only 4 non-jetlagged days.
Get a guidebook specifically for London; Rick's is good and very detailed, with a lot of excellent logistical advice that will help you be efficient sightseers. Make a list of places you'd both like to see. If you don't seem to have enough common goals you can split up for part of the time to focus on individual interests. Think about what places you might want to visit only if you're lucky with the weather; stay as flexible as you can, except for sights needing advance-purchased tickets.
If you enjoy walking and the weather's decent, the just-show-up 2-hour walking tours offered by LondonWslks are a fabulous value at £10. There are dozens and dozens of them. The winter schedule won't be available until this fall, but you can take a look now at the summer offerings, many of which will be offered during your timeframe.
Come back here with your list of potential sights and we'll warn you about tickets needed ahead of time, etc. Your list may also suggest that it would be good to stay in a particular part of town, though it's probably more likely that your sightseeing targets will be scattered.
You can use booking.com for a preliminary idea about lodging costs; the reviews are relatively trustworthy, and you can go to the hotel's website to make your reservation if you prefer to do so.
I find the Premier Inns generally reliable (check rates on the corporate website, premierinn.co.uk) and used them heavily this summer when I had a roommate to share the cost (as you do) and wanted to be sure of air conditioning (which you certainly will not need). Outside of air conditioning season, there will be less-expensive options, but recognize that lodging costs in London are very, very high, so it will probably seem that you're not getting much for your money. Do look at photos on booking.com (and maybe also TripAdvisor) so you know what you're getting--often a tiny room with an even tinier bathroom and charmless, dorm-like decor. There are still some hotels in London with bathrooms down the hall. They are less expensive. If you want to avoid those, the term to look for is "en suite bath".