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Yeah, they get stuck. I've been able to get some out and others not. Most carbs (older Dellorto, Mikuni, Amal) don't have o-rings to seal the idle screws, instead they rely on the fine threads sealing so I didn't worry about the old o-ring too much. Apologies for stating the obvious - the area between the tapered tip of the screw and the tapered seat meters the fuel. Any air leaking down the threads of the idle screw can easily be accommodated by turning the screw out a little more. It least that's how I handled it...
I straighten a paper clip out and put a very tiny [1mm?] 90* bend on the end with pliers and then use it to "fish" such things out.
Can you tell if it is rubber or metal? If rubber, maybe a little heat to soften it I keep a set of cheap dental type picks around for these jobs.
Rebuild video: http://tinyurl.com/4syt574nLooks like you have the body of a jet broken off in the casting. That should be a threaded hole.The video (in German, sorry) gives nultople quick views of the bare body of the carb. If you scrape the hole a bit you should see brass. Maybe Kroil, heat and an EZ-Out might clear the passage.