If you are staying at the place Eva recommends, great. Otherwise, be very careful. Airbnb does not have a particularly good reputation in London where there are very few genuine bed and breakfast establishments. A lot of people have jumped on this bandwagon offer ending rooms in their homes, but this is totally unregulated. You may be very lucky and get a truly genuine family who give you a fabulous experience, but you may also get a totally unforgettable experience for totally different reasons. My son lives in London and says this is often in local press. If somewhere seems too cheap, it is probably for a good reason. Buyer beware. The Premier Inn, County Hall, right by the London Eye apparently does a decent level of accommodation at cheapest prices you are likely to get., it's a well known British chain. Very central location.
Good luck.