Like David, I too have owned both and still have the K13S.
I wouldn't use a K12 for the London commute. As others have said, many are faultless but mine was a pig in fits and starts and would, had I been paying a dealer to keep it running for daily use, would have cost me more than I paid for it.
The K13 is a MUCH better bike. However, neither is the perfect commuter, depending on the length and type of your commute, from where and to where. There are much more reliable and less costly-to-run bikes out there. They won't hold a candle to a K on the sports/touring front but you are not doing sports or touring. Also, David's bikes are paid for by his company which offsets some of his running costs.
If I were in the market for a commuter, a decent 600 with weather protection and a great topbox would be my choice. Something boring and bland from the far east which will not cost a lot to run, won't set fire to your trousers but which will go on and on, day after day.
Oh and welcome to the madhouse that is EuroKClub!