David,
Pretty sound advice so far regarding your trip this way.
For what its worth, I'd suggest staying in England and travelling back this way another time to experience Ireland or Scotland, and I do love Scotland, so its not just the Englishman in me talking!
I would suggest that even if you aren't playing golf every day, there is more than enough opportunity to mix it up a little. You could easily spend 10 days playing the London heathlands and not get bored, but it would be pretty easy to make a trip or two further afield?
Walton Heath, Sunningdale, Swinley, The Berkshire are all classics and you could easily add New Zealand, St Georges Hill, Woking or many others to that list of those close by. The Addington is a little more central, south London way, and though I've not played it myself, certainly sounds worth it! A handful of those should form the basis of your London / heathland golf.
Pushing a little further out, Royal Ashdown Forest is probably worth a look, Pulborough and Liphook also. If you fancied a trip to Oxford or Cambridge, then Huntercombe or Royal Worlington come into the equation.
Further afield still, as has already been mentioned there is Royal Cinque Ports in Deal (take up Tony's offer, and also if you wanted to play Royal St Georges a night at the B&B Mr Chaplin mentions shouldn't be so quickly dismissed!), and then of course Rye is sort of that way also.
Garland's trip to the south west is also possible for several days and probably a good way to get away from London. Check out Painswick if you fancy a bit of quirk, and if Sean can't get diaries to work out at Burnham & Berrow, I might be able to?
I think there is more than enough to be getting on with, but if you did fancy a little trip north, to the Midlands (just under 2 hours drive from London) I'd be happy to host at Notts (Hollinwell) which you could probably tie in with say Woodhall Spa or Beau Desert, assuming we can get diaries to work out?
Cheers,
James