Having just returned from JFK-LHR-ORD-MIA-LGA, I have some tips!
Definitely use the AC in MIA. The terminal is horrible -- surly staff and almost no announcements made, but the AC makes up for it by being quiet, spacious and calming with very helpful staff.
I've never had to stay in a hotel in London, but $46 sounds amazing given the strength of the pound: cheapest youth hostel in central london I heard of five yrs ago when inquiring for a friend was 20pounds.
It's fordidden to BYOB, and the AA transatlantic coach food is nasty IMHO. Interestingly, they served the pizza snack and not the (even nastier) snack box on my LHR-ORD flight, but I'm still not a fan! I recommend you go to Selfridges Food Hall and stock up on some nice things, or even the Sainsbury's in Paddington if you're taking the Heathrow Express train (but might not be open very early AM). The LHR AC had some nice salted nuts and shortbread biscuits but not much else. And it has a free bar.
Don't forget the Park Avenue checkin for elites at LHR -- across the street from the main Terminal 3 dept, but if you're coming by the train you might find it easier to use the in-terminal 1st class desks.
Concourse D in Miami airport had nasty looking expensive food and $3 bottles of water, so bring your own from New York instead or enjoy the snacks at the AC.
You won't get LHR arrivals lounge access (and access to the T3 AC for arriving passengers is now blocked off) with a coach MIA-LHR ticket but could pay to use the arrivals lounge -- $35 I think for coach passengers. Never tried it myself.
Dunno about US plugs. I expect not -- why not buy a cheap adaptor before you travel?
Some London tips: Hampstead Heath is pretty at this time of year and good to walk off your jetlag. You could have lunch in Hampstead nearby too. Also I recommend the current Tate Modern exhitibition of Henri Rousseau if you like that sort of thing...