Paul, I am sure you will be very happy with your new TS55. It has a dust port made to fit both the Festool 27mm and 36mm hoses. The smaller rubber tip of the 27mm hose goes inside the dust port, while the larger 36mm tip goes over the outside of the dust port. The vac end of those Festool hoses are termed "50mm" by Festool. Actually they conform to the international standard for shop vac connectors.
I have no personal experience with the Karcher vac. If it conforms to the international standard, probably the easiest approach would be to buy a Festool hose. With a Festool CT dust extractor I always recommend using anti-static hoses, but unless your Karcher vac supports the anti-static feature, you could use a Festool non-AS hose. The collection from your TS55 will be improved by use of a 36mm hose.
People have found fittings between the TS55 dust port and standard shop vac hoses. Just remember that any hose heavier and less flexible than a Festool AS 36mm is interfere with the balance of the TS55. A lot of people who tried using the adapter with the large shop vac hose eventually bought a Festool hose.