G'day mate,
Don't know if you are continuing down this path or not but I have looked into this motor in great detail, as it's the one in my Lotus Elise.
There are at least two different versions of an updated thermostat kit been made for these motors. The pick of these is called a "pressure relief thermostat", I believe this is the style that Landrover are issuing. Yes you will need to remove the old stat from its housing. (if it's the same housing as the Elise that's just an open the housing and remove job)
I know of examples which are still quite happy after 150+ thousand km, but others which have failed in much shorter times. The main difference seems to be those who have good longevity have always let the motor warm up properly before giving it the beans.
Modes of failure of the K series motors do vary, at least in the Elise. Most have had heads "walk" because of the plastic dowels (headgasket blows by). Liners dropping is most common in motors which have been rebuilt, and liner highets not checked and adjusted correctly (should stand proud by a specfic amount) Another common issue which the head gasket gets incorrectly blamed for is the inlet manifold gasket leaking, allowing coolant to mix with oil, certainly worth checking if you have those symptoms.
Crankshaft failures get reported as well, but usually only in motors being reved hard with head work (7000 seems to be all people can get away with and remain fairly reliable)
The upshot is that they are a fairly reliable motor if looked after. When and if the dreaded HGF comes along, rebuild with MSL head gasket and steel head studs, taking care to set the liner heights correctly and job's a carrot!
Have fun!
