Another good tyre to try is the Dunlop Roadsmart 3. These have stiff sidewalls as standard and support the bike well. I have them on my K16GT and they are lasting well. Just done a fast run down to the Alps for a few days, no instability, good in the wet and steer very neutral. Sat at over 100mph for large chunks of the journey, rode it hard over mountain passes, and they still look good after nearly 3,500 miles which is a lot on this bike.
Should suit a K13 quite well.