Flights from Belfast to London are as follows:
From Belfast International:
easyJet to Gatwick, Luton & Stansted
From Belfast City:
Aer Lingus to Gatwick & Heathrow
BA to Heathrow
FlyBe to London City
No London airport is close enough to the city centre to be able to take a taxi without incurring massive expense. Heathrow is 17 miles out (but heavy traffic can often mean that the journey takes over an hour). Gatwick is 30 miles out (and likely to take nearly an hour and a half in a cab because you'd be coming in via south London). Luton is 34 miles out (and will take well over an hour in a cab). Stansted is 40 miles away from central London (and, again, well over an hour away by taxi). London City is just 9 miles away but it would still take at least half an hour in a taxi.
London's cabs are some of the most expensive in the world. You'd almost certainly have to pay well over £100 from any airport except London City (and probably over £50 from there). I've known people who've been charged (the correct meter fare) of over £200 from Heathrow after they've got stuck in traffic. If you tried to save on the fare by using London City airport you'd almost certainly have to pay far more for your flight (because that airport serves mainly business passengers).
So you should take the train, from your arrival airport, into central London and then use the Underground. (If you arrive at Heathrow you can use the Underground all the way. If you use London City you can take the Docklands Light Railway and the Underground).
Most of the more reasonably-priced accommodation in London is some way out from the centre of the city. (So you might need to consider staying a short Underground ride away from the places you want to visit). However several ABers have recommended the Strand Palace Hotel which, while not particularly cheap, costs nowhere near to the £500+ per night that many of the posher hotels charge. It's directly opposite the Savoy Theatre:
https://www.strandpalacehotel.co.uk/