I must agree that you have found a beautiful machine in remarkable condition.
According to the London Douglas club records the frame number places the bike as 1920/21. It is a 2 3/4HP (348cc) probably a model W, The gearbox should also have a number, this will help to identify exactly which model it started out life as!
Sadly the wheels are not easily found but can be built up from spares that can be obtained some parts through the Club others from specialist wheel builders.
Engines do become available from time to time, the frame you have should carry an engine with the numbers in the range 47000 to 49000 but engines from a wide range of 2 3/4's will fit. Douglas did not match the engine numbers to the frame numbers.
I suggest that you consider joining the London Douglas Club, we have members all over Europe and the rest of the world. Should you wish to do so please check with the Club website where you will find information on the Club. douglasmcc.co.uk
If you have any problems feel free to email me at my address here on the site.