Autumn/Winter in London

Hi. We are two friendly, respectful, clean and tidy, easy-going people (a sixth form College teacher working in East London and an HR Director working between universities in the South-East and London). I am looking for somewhere where I can stay on 2 or 3 nights of the week during the Autumn and Winter (to cut out a commute from Kent, where we live) and then we would both stay over at weekends from time to time, although almost certainly not every weekend. Our preferred area would be south of the Thames probably, perhaps Blackheath or Greenwich, or alternatively around Westminster or Pimlico (close to Victoria train station). But we would consider anywhere if it seemed suitable. Our adult children live in other parts of England so being in London is more accessible for us all to meet up from time to time as well. We didn’t know what rent to put so the price we’ve put is just to fill that box (the photo was taken at our son’s wedding in June).