Another option would be to rent a room in a university accommodation. Rooms that are usually occupied by Bachelor students become vacant at the end of May/beginning of June, when the second term ends and the summer holidays begin. At which point universities try to rent these out to summer schools, language course students, and all those people who need a room for the summer in London.
For example, Imperial College London rents halls out for the summer. Some of the halls are in Zones 1 and 2. Moreover, the price range is likely fit in your budget. It will be up to you to look for options.