Connor's kinsmen killed The Kurgan
Not that he stayed dead.
I can't count how many times I've seen Highlander, and it never occurred to me how bad Dougal and Angus MacLeod must have messed up The Kurgan after he stabbed Connor.
You only see The Kurgan getting swept away by the battle, which is good because seeing him get carved up at the beginning of the movie would have weakened his aura of invincibility.
If you don't know history, you don't know anything. You are a leaf that doesn't know it's part of a tree.