London Hotel or Flat - Budget & Location are both important

I'd recommend searching on booking.com, which has good filters and reviews by people who have actually stayed in the hotels. You can tailor your search to a price range. I think you'll have few choices in the $100 range unless hostels are OK. You might also see what other Premier Inns you could find.

Being near a station can be convenient, but your day trip destinations are each served by a different station. There should be good tube connections all over the city though. You might be tempted by a hotel in the suburbs, but consider the time and cost of traveling in and out of the city each day, and the convenience of a centrally located hotel where you can take a midday break, nap or whatever.

Thanks for pointing me in the direction of booking.com - I had not thought of them in relation to England. No suburbs for me - best to be near the action.

You might be able to get a room for 100 Gbps, at the Celtic Hotel . It's right by Russell Square . Clean and comfy , some rooms have private faculties other cheaper rooms share ( but share washrooms are very clean and you only share with 2-3 rooms max ) . Rate includes a full cook to order breakfast !

I've had great luck with Airbnb. Last year I rented a three bedroom flat in London for ~$150/night. You could easily find something within your budget. If a train station is important to you, be sure you know which one before you select your lodging. There are several in London.

Look at Cherry Court. It is 75 British pounds per night, so very close to $100. It is a couple blocks from Victoria Station. The room we stayed in was small but worked fine for us. Included is a basket with fruit, cereal bars, etc.

Try London School of Economics. They have several campuses in London and you may be looking early enough the you have the pick of locations. I haven’t stayed there personally but my son did at the High Holborn location and he was happy with it. I’m staying at Cherry Court Hotel in May and paying £75 a night for a double room. It may be a little tight for 2 people but it’s just a place to sleep and shower, right?

100 US dollars is about 72 British Pounds, that's a very tight budget for any big city, much less one of the most expensive cities in the world. Sunday night prices in London tend to be inexpensive, so including two Sundays in your trip will help. Unfortunately, Saturday night prices will tend to rip the seams off your wallet.

I'd suggest upping the budget or, if that's not possible, then either consider postponing the trip or consider visiting other cities in the UK where prices are much lower.

I'd also suggest to start thinking in Pounds. It'll make the trip go smoother as you'll be able to quickly determine how expensive prices are at stores, pubs, restaurants, and cafes without having to convert everything to dollars.