Maybe you need another spacer. Try removing the top cap and bolt and putting a 5mm spacer on top of the stem, then add the top cap and bolt again. Loosen the stem and lightly tighten the top bolt until you have enough preload that the headset spacers don't turn by hand anymore. Now tighten your stem again.

Any good?

Not all headset bearings use the same contact angle for the crown race and bearing. 36° and 45° are the two common angles. I don't know if Chris King uses either of those, or something of his own choosing. But if you have a mismatch, it may not ever work quite the way it's supposed to. In that case, I'd measure the bearing's contact angle and pick a crown race that fits it.