Budget: 200 £
Hazlitt's is very central and is a lovely, small boutique hotel. Some rooms have four poster beds and there is a small library with some first edition books in it. It's in Soho so perfect for the theatres and the West End. There are also very good pubs and restaurants nearby. The 'regular' price is a bit over your budget but you may get a deal.
It depends when you're going, but central London hotels, especially those that offer a bit of ambience to them are probably well over your budget. If you'd like something fun and funky, I'd suggest citizenM hotel London Bankside which is just down the road from the London Bridge Tube station and, from there, you can get anywhere you want to go in London in about 20 minutes. There's no view from the hotel and the rooms are small (pretty much any hotel you're looking at in this price range will be small) but they're fitted out amazingly well and have free wi-fi. You can also set your own 'mood lighting'. The hotel is also back from the main road so it's quiet to sleep. I loved it there.