When I arrive at Heathrow, I take the tube to the Victoria Station area, where there are plenty of reasonably priced hotels. I usually just look for one on www.booking.com , where you can enter "London Victoria" as your destination. If you pay attention to the guest ratings and reviews, you should have no problem.

The last time we were in London, we stayed at the Lidos Hotel, an easy walk from Victoria Station. It should be in your price range, but there are plenty of others like it.

Victoria Station is within an easy walking distance to Westminster Abbey, Buckingham Palace, and several parks. It also is a major transportation hub, making it convenient to get to any other place you might want to visit.

The OP needs to catch a train from Victoria so Marble Arch, the Strand Palace, Citizen M et al aren't very convenient. Since they can stay at the Best Western for free that would seem to tick all the boxes.

BTW -- if you do decide to take the tube in from LHR, get off at Hammersmith, cross the platform - only about 30 feet w/ no steps - and get on the next train which will be a District line train to Victoria.

Check out the Thistle chain. They are respectable but not fancy and I seem to remember getting a really good deal on one next to the National Gallery a number of years ago. Just randomly checking for two weeks from now it looks like rooms run 150-170 pounds at some locations.

I stayed at a Grange hotel (small boutique local chain) near the British Museum in February and it wasn't an arm and a leg. I can't remember the exact price, I would have to look it up, but it was probably something like 160 pounds a night.

My most recent trip, in September for the wkd, I stayed at the Waldorf London (which is not a Waldorf Astoria property, but very nice nonetheless) and that worked out to 400 bucks a night for a large deluxe room with huge bathroom (double vanities, even). Which was QUITE nice and in a perfect location.