This dataset from the Valuation Office Agency measures the total number of homes. The VOA collects various data about each property including council tax band, year property was built, and property type (eg house/flat and number of bedrooms).
The visualisation below shows the number of homes in 2015, the change in number of homes since the year before, and the percentage of these homes belonging to aggregated council tax band groups.
The data behind this is available on the London Datastore.