You can't really get a hotel close to good sites in London and Heathrow airport... its either/or as Heathrow is on the outskirts of West London.
As a suggestion... Travelodge hotel in Marylebone..this is right opposite the tube station, very cheap - I'd be surprised if you pay more than $75 (obv don't expect much more than a bed and small tv!)..you can get trains from there to Heathrow and to sites pretty easily
Yep, Travelodge is the way to go. They've got hotels scattered around London, all rooms are in the $75-$150 range although you do get some special deals running every now and then. Basic rooms but decent enough for a night or two. Check their website.
