Well I must admit, I would hit it, but scientifically… but first.
Someone mentioned it before but there must (may be) a couple of ejector slots, and the proper wedge ejector is usually pretty effective. Try tapping (NOT beasting) the taper drift in. Once its firm, knock it in the ejecting direction.
If you havent got ejector slots, then Id use Kwils thingy to get tension on, lock the quill, and use hte spanner flats to stop it rotating while i tensioned up. Then I’d use an ali or brass drift down the spindle. Tap it – don’t beast it. Like shifting any taper, if oyu can get good tension on, a tap usually works. But be mindful – even with a locked quill – you are hitting your bearings.
If that doesn’t work, I’d try to tighten Kwills thing a bit more and put more tension on, press a hammer agin the mouth of hte sleeve, and strike(!!) the other side sharply .
If that doesn’t work, you have a problem.
BTW are you sure yu have KWILS device right? You have a bolt or somesuch that you drop down the spindle. Its falnge or head fits on whats jammed. The shank sticks out below the bottom of the quill or jammed sleeve. You have a hollow “Washer” with a couple of spanner flats on it.(or a pair of tommy bar holes) It fis OVER the jammed sleeve and bears on the body of the quill. You put a nut on the shank and tighten up to apply tension in the freeing direction. How do you need to hold the quill? You are not tightening onto the quill but againsit it, and the reaction is between the bolt/withdrawing bar and the “washer” or withdrawing sleeve.
Are you sure you haven’t missed a trick, because that kind of extractor sleeve is normally pretty darn effective.