I would suggest that this is a relatively involved repair, and premise my recommendation with this:
If you aren't comfortable using (or you don't/won't have) a shop manual, all the standard wrenches and tools needed, or remembering and paying attention to all the details involved in engine repairs, I wouldn't attempt this repair yourself. But by all means, if you want to learn, do it!
Sludge in your cooling system is most likely oil in your coolant system. The key places where oil and coolant interface and are close together in the engine is:
- water pump
- oil cooler
- head gasket/cylinder head
I would say the stator cover damage is likely unrelated, but definitely something to investigate. I think on the K6 the stator is nearby the water pump but I might be wrong. Take it off and check things out. The stator cover isn't something to just patch up, the cover itself has the stator bolted to it. If the cover is damaged or cracked enough, the stator inside won't be sitting in the right position. Definitely replace or upgrade to a Woodcraft cover.
So I would start by draining the oil and coolant, and checking the fluids for weird colors, textures, debris, smells, etc. Check if the oil is milky or frothy, and see what the coolant looks like. Then I would check the water pump for dripping from the "weep hole". Common practice, google that inspection if you are not sure how to do it. That would tell you if your mechanical seal in the water pump is potentially leaking, and help you narrow down your issue.
The oil cooler is super unlikely to leak like this, since it's solid state and has no moving parts. But worth taking a look at.
Head gasket is the worst thing on the list for sure to have an issue with. I would resort to this last. If you know the water pump, hoses, and oil cooler are good, then that's the only remaining likely problem area. Look around your engine for signs of leaking of any fluids. And if you are equipped, take the cylinder head off and check it out.
Hope this helps, if this is too much, then I would just bring it to a shop.
-Mike