Hey!
I am a current PhD student at LSE. I am planning to move out of my accommodation to live with my partner.
If you study in a UOL university, I have an accommodation available in the International hall of UOL! The rent is 266, it is located in central London (Russel Square) and is a 17 minutes walk from LSE, KCL, UCL, and SOAS. Also, meals are included! Plus the tenancy is till 22/06 which matches your criteria perfectly.
Due to the central location, and savings on food expenses, I feel it is a great deal. Message me privately if you are interested!!