The Churchill War Rooms are quite expensive if you only want to see the war rooms themselves and ignore the Churchill Museum. If you want a full visit, it can easily take half a day or longer. To avoid a potentially very long ticket-purchase line (and conceivably a sell-out), you must buy timed tickets in advance. This will not work well with your limited time. You can't be sure when you'll get out of LHR, into the city, drop your bags off and finally get to your first sightseeing stop. Although this is a quality historical site, I would not try to squeeze it into your very tight schedule.
The British Museum is a monster. It takes multiple visits totaling multiple full days to see the whole thing. The best you can do is look at the website ahead of time and select a very few areas to prioritize. The ground floor tends to be extremely crowded, including the highly popular Egyptian section. You have to go through a security line to enter the museum (possibly faster if you arrive with no purse, bag, etc., but I'm not positive about that). I've stood in that line for more than 30 minutes on several occasions, so check immediately to see whether you can get a timed entry online that works with your schedule. You really cannot afford to spend much time waiting in line.
I've not been inside St. Paul's and visited the Tower of London a very long time ago, so I have no comment on the time you're likely to want to spend there, but the Tower of London would normally be a significantly longer visit.
I must warn you that $150 (which is just £111) is a very low nightly hotel budget when you need space for four people. You'll need to select carefully to avoid places with really low reviews and/or inconvenient locations. If you stay very far from the center of town, you'll burn precious time commuting back and forth to sights (you'll be doing that four times within about 24 hours), and Underground fares are not cheap. A look at booking.com will give you an idea of costs. Booking directly is often somewhat less expensive.
Premier Inn hotels are often recommended, and they are typically very reliable. You can check their room rates on premierinn.com. Be very careful about the type of reservation you book; the lowest rates are not changeable or refundable. Flexible rates will cost substantially more. Up to two kids eat breakfast free with an adult paying for a full (not continental) breakfast, but the kids must be under 16.