One suggestion for looking for cheaper housing would be to look for things a little further out from the center but on a train line that would make it east to walk to the Strand from Waterloo Station. More central, some people who were at King's (so next door) lived near London Bridge, Paddington, Canary Wharf, Whitechapel, and Camden (for some potential places to look). Thinking about convenient bus routes for when the trains and tube stops running might be useful.
Public transport can be annoying and a pain sometimes, especially when there are strikes but especially since you're familiar with commuting and navigating in a big city hopefully it shouldn't be too much of a drag.