I don't think you'll find Beckenham to be falling over with Hotels & B&B's, it's commuter land and doesn't have the call for them, on the other hand, from memory, it's only about 20mins. by train from London, Charing Cross. I originate from that part of the world.
From Charing Cross to Kings Cross, where you pick up the train for Cambridge, is a direct tube connection on the Northern Line.
You may well be better staying in London, but I did find these: http://www.hotels.uk.com/beckenham.htm