How to remove a screw driver bit that is stuck in the chuck of the drill motor

The fact it broke off means the screw is probably very stuck. I would first use a penetrating oil (though WD-40 might also work), and let it sit for awhile, before trying anything further.

Hindsight is 20/20 of course, but it would have been good to do this first. If you see any corrosion or rust, or suspect it's needed and have time, this never hurts to do.

You may be able to use a small drill bit to extract the broken extraction bit, and then try again -- but be careful as you may damage the screw beyond my next tip..

If you can't get the tip out (or don't want to risk trying), I would probably use a dremel cutting wheel, and basically turn the screw into a slotted screw. This can be difficult -- not deep enough, and you'll just strip the screw worse trying to get it out, and too deep and you'll weaken the head so much it'll probably snap off when you try to turn it.

The next thing beyond this is to use a "reverse drill bit" or "left-handed bit". You can drill right into the middle of the screw itself (with your drill in reverse), and the bit should actually come out before you've gotten all the way into it. Use a bit that is smaller than the thread size, otherwise you'll strip the threads so you won't be able to put another screw back in. Again, judgment is necessary here, too small and it won't get enough grip, too big and you'll strip the threads -- though maybe by this point you don't care that much :)

The next step secret to getting more of this faucet apart are the two slots cut across the cupped part. See the arrows in the following picture.

It looks to me as that the cupped part is threaded down inside the larger diameter part that sticks up from the hole in the sink. There is probably a special tool used to engage these slots to thread the part out. It may be possible to lay a piece of metal bar stock across these slots as a tool. If you do that make sure that it has nice square corners.