All credit to the original I-BMW poster who posted pics of the whole assembly. I've just completed a rebuild but I have some reservations. First though, I was seriously looking at HyperPro 'cos it was adjustable but cost said I should justify it, considering mine has been shot for ages.
With the stock damper I read handling is heavier on hard twisties - I'm not on twisties that often throwing my weight around so I wouldn't know. Next a consensus says the stock damper stops low speed wobble - yes I might believe that now I know how it works from the inside.
The stock damper uses two pistons which can slide a small amount back and forth. i.e there is absolutely no damping for small steering movements as much as 10 degrees or more - so most of the time it doesn't do a lot. Once the steering angle increases you get some damping until the internal damper pressure equalises - then you get another 10 degrees of no damper action. So the first turn into a bend gives you damping, but unless you put on more steering, you're soon back to 10 degrees of no damper action. I'm convinced the stock damper is purely to stop oscillation. Now the HyperPro (and possibly Ohlins) has a variable action so it works different to stock and its purpose is more for handling. I haven't seen a negative post about the K's handling - except it's heavy.
All credit to the I-BMW archives for info and photos on damper rebuild, but there are pitfalls. The oil used is a critical part (nobody has really confirmed its viscosity) and in my case, the small piston bypass holes were partially blocked. Also there must be no 'nicks' in the chromes rods passing through the seals. After re-assembly with 10Wt fork oil, aggressive turning of the steering on the stand blew out a seal and oil - very messy!
I'm now trying a lighter oil, the damper action is softer and the seals are holding. I'm researching the original KACO seals for their spec. I've got reservations about the ability of the stock design to hold in oil, the lack of boots or gaiters on the exposed pushrods, the temperatures in front of the radiator in slow traffic to give good service life, and the limited travel distance requiring high internal pressure during damping - but these may be reasons why they don't last that long!
If you are looking at buying a bike and need to check the damper is both working and durable, sit a passenger on the back to raise the front wheel and from the front turn the steering back and forth hard. No resistance = no damper, a load of oil in your face or on the fender is a dead damper!