Flying there will be the cheap part of your trip.
Be prepared to spend a lot of money on hotel and food.
I've stayed at Columbia Hotel at Lancaster Gate twice, and it's very clean and safe for the price. It's definitely not new and luxurious, but the money you'll save will be worth it. Plus, there's a yummy free breakfast.
Things worth doing:
Tower of London (worth the money)
Westminster Abbey
Hampton Court Palace (30 min train ride out of the city, but so interesting)
British Museum
Oxford Street (shopping in stores we don't have here)
National Gallery
St Paul's Cathedral
See a show at West End (you can buy discounted tickets the morning of at the TKTS booth in Leicester Square)
If you have time:
Kensington Palace
Changing of the guard (get there early or don't go)
National Portrait Gallery
Portobello Rd Market (Saturday morning in Notting Hill)
See a play at the Globe Theater
Overrated (just my personal opinion):
London Bridge Experience (ended up being a haunted house)
Tower Bridge
Picadilly Circus
Texas Embassy
And a few reminders:
Londoners make dinner reservations, so be prepared to be turned away from nicer restaurants or wait a long time if you haven't booked a table
Buy a pass for the Underground (unlimited use per day) for Zones 1-2 and you will likely save money on tube fare
Take the Paddington Express train from Heathrow and then a cab to your hotel - it's worth it if you have lots of bags. Tube stations do not all have elevators/escalators and lugging suitcases up stairs is no fun.