Err....you say there is no tenancy agreement so when did the tenancy actually start? - this is important as you will need proof in order for your S.21 Notice to be effective. I think you might be better to go the S.8 Grounds 8,10 & 11 as this does not rely upon the date at the end of a rental period as does a S.21. No tenancy agreement means you have a periodic assured shorthold tenancy from the start, and determined by the frequency rent is paid i.e. probably monthly, yes? If you reply I might take a day or two before I can look at it.
The advice I give should not be construed as a definitive answer, and is without prejudice or liability. You are advised to consult a specialist solicitor or other person of equal legal standing.