I bought a set of carbs with the slow jet broken off as you describe.
I used a small hand drill to make a shallow pocket in the jet and a small easy out to twist out the remains. Select a drill bit that is only just larger than the easy out.
Given the experience, I believe one of those left cutting drill bits would have extracted it in one motion. Those can be hard to find, though.
Gordon's clever trick probably would have worked too. However, the brass is soft and the tapered Phillips point "might" expand the soft brass outward against the threads in the body. I remember when I initially planned the extraction procedure, that a driven screwdriver might just be enough to turn out the remains. A had a small tipped flat blade screwdriver that I was going to slowly tap into the brass. But, then I found that my smallest easyout (even with a broken tip) was still small enough to work without knarling the carb body threads.
Even with the flat blade screwdriver, I still would have drilled a shallow hole slightly narrower than the screwdriver tip.
Lastly, I have an optvisor with a 3x magnifier which gets up close and personal with these small bits of trouble. It really makes you aware of just how tools interface with components you are working on. Yeah, they look stupid on your head. But, many a giggler has been found wearing one many times after the first try. They are that good as tools.
Good luck! Good patience! And, remember that many problems can be solved with a suitable application of explosives. But, not ALL problems are best dealt with in that way.
Cheers,
