I was building a mount for my custom bobber. My 1/4" carbide drill bit broke off into 3/4" plate of steel.
How do I remove it?
If you have a welder.
Weld a metal rod to the top of the drill bit at a 90 degree angle from the drill bit.
You can then use the rod as a lever to turn out the drill bit in reverse. You may want to take a hammer and punch and hammer on the end of the drill bit a few times to loosen it.
How much of that drill bit sticking out?
IMO opinion, even if it a bit of welding splash gets on the plate, you can grind it off. Looks like a work in progress. If a piece of the bit happens to get welded to the plate, you can grind it a bit to ensure it's free.
I've used the trick on very large snapped bolts but never a drill bit and am unsure of whether or not the bit can be welded to the rod in my scenario.
The drill bit has flutes (grooves in the sides). Try driving nails into both flutes, then grab with locking pliers and turn. Lubricate with penetrating oil first to help.
On second thought- instead of nails, which are generally very soft steel, use tougher steel pins... the easiest of which to find is the shank of a dull drill bit, everyone has some dull 1/8" bits lying around. You can cut off the fluted part, or just use them as is.
For go the previous answers! I used to do it all the time as a Big truck Mechanic. use two lite hammers and two punches half the diameter of the drill flutes(grooves on side of the "drill bit"). Ask a friend to operate one set of hammer and punch you the other set. using the drill flutes You both set punch at opposite directions,close to 90degree angle to broken bit point in opposite direction to original drill operation. Now both of you tap punch with hammer gently (hard hit not needed)try to tap as close to same time as you can. While timing isn't critical it does help! The time you take to gather the tools will take longer than removing the bit! It will screw out backward in about 30 seconds to one and a half minutes. It works absolutely every time. "CB"handle is "CookieMonster"just relax and enjoy the info.
Ideally, with a heavy duty screw/bolt extractor. If that does not work, here is a helpful walk through I recently came across about how to remove a broken but from an engine block: http://m.lowes.com/pd/SpeedOut-4-Piece-Screw-Extractor-Set/50217437
If the bit isn't well jammed into the hole you might be able to get the tips of some needle nose pliers down the flutes and grab hold of it with that. If you can get hold of it, unscrew it out of the hole. This has worked for me but the bit in question wasn't tight in the hole.