Broke off carbide drill in hole.

My understanding is that files are usually fairly soft to facilitate cutting the teeth and later case hardened. Is hitting a carbide drill fragment with a soft-ish steel punch going to work? Carbide is brittle so it might work but I have my doubts. I would start with a standard punch. If you're in a desperate situation, make a punch, harden it with a water quench and temper
That may be true for newer manufactured files. It might explain why the newer Nicholson files re junk compared to the files from thirty years ago.

I have been making center punches and prick punches from files for decades. They are definitely high carbon throughout. A spark test of the central core will show that. Another indication is that the file will snap in two if bent.

I wouldn't use a square ended punch as the objective is to fracture the carbide. By driving the point of the punch into one of the flutes, you have a better chance of fracturing the drill. At a 2.5mm diameter, there isn't much room to work but with patience it is possible to clear the hole.
 
An edm will blast right through that carbide drill.
We used an edm in our Speed Shop/Machine shop. It needed a wax or whatever dam built, around the broken part, to hold a lake of Diesel fuel. Once you had a puddle of Diesel fluid surrounding the broken bit, you erected a rude and crude hand drill tower creation above the broken bit, hooked the working carbon "drill bit" to a power unit like a buzz box welder, and turned your drill creation to feed the carbon into the broken part. You wore an arc welding helmet, and observed how much the carbon arc was de-solving/eroding the broken part. You turned the crude hand drill down more and more into the broken part until it was gone or at least until you got through the hard stuff. It worked so well, that whomever burrowed it (as a "buddy, buddy" favor) decided to make it theirs rather than ours. You could probable scab your own version together out of stuff you already have. Just keep the arc in the diesel fuel and go at it. You can even use a hex shaped carbon, and then use an allen wrench inserted into the hex shaped hole you eroded out of the work piece, to unscrew a broken part that is flush or below surface. You young guys probably never heard of carbon arc "welders" which actually were more like brazing torches than steel welders. The carbon arc torch was just a mechanism that allowed you to touch and then separate two carbon rods and allowed you to make some intense heat. It was not very directional, but it worked fine with a flux coated brass or bronze rod. You could also heat "to bend" metal as well. You had to wear an arc welding helmet. The carbon arc torch worked with the old buzz box welders, just the same as the recommended edm Batmanacw is talking about.
 
We used an edm in our Speed Shop/Machine shop. It needed a wax or whatever dam built, around the broken part, to hold a lake of Diesel fuel. Once you had a puddle of Diesel fluid surrounding the broken bit, you erected a rude and crude hand drill tower creation above the broken bit, hooked the working carbon "drill bit" to a power unit like a buzz box welder, and turned your drill creation to feed the carbon into the broken part. You wore an arc welding helmet, and observed how much the carbon arc was de-solving/eroding the broken part. You turned the crude hand drill down more and more into the broken part until it was gone or at least until you got through the hard stuff. It worked so well, that whomever burrowed it (as a "buddy, buddy" favor) decided to make it theirs rather than ours. You could probable scab your own version together out of stuff you already have. Just keep the arc in the diesel fuel and go at it. You can even use a hex shaped carbon, and then use an allen wrench inserted into the hex shaped hole you eroded out of the work piece, to unscrew a broken part that is flush or below surface. You young guys probably never heard of carbon arc "welders" which actually were more like brazing torches than steel welders. The carbon arc torch was just a mechanism that allowed you to touch and then separate two carbon rods and allowed you to make some intense heat. It was not very directional, but it worked fine with a flux coated brass or bronze rod. You could also heat "to bend" metal as well. You had to wear an arc welding helmet. The carbon arc torch worked with the old buzz box welders, just the same as the recommended edm Batmanacw is talking about.

The edm I'm talking about is nothing like you describe. The arcing is tiny. It runs in a fluid bath and it crackles and pops but no welders involved.
 
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