Integrated type headset has zero stack height on the bottom,
and in the case of my cane creek one, just a squat ring above the top bearing.
Because the head tube is machined to hold the bearings within it's length..
[AFAIK] Chris King doesn't make an integrated headset,
they machine the aluminum (etc) bearing cups to press onto the frame.. externally .


so your fork's steerer tube is cut too short?