Before starting to drill another mistake/hole lets touch base on drilling small holes in steel. You should be no less than in a drill press and with the piece locked in place. Using proper speed for this size drill should be at least 3,000 rpm and the most important trick is drill to the stop. Meaning adjust your stop to a just touch, raise drill, turn stop a good .010 and drill. Get in the rhythm and just keep repeating, takes no time at all. You did not do this and when it was just ready to break thru the drill jump forward and hooked in. You well not get yourself in this trouble if you follow the above.
Since the drill is not thru enough to push back with a dentist tool. Then you probably need to heat end of drill with propane torch and attempt to drill with same size drill. You only need to heat .020 -.030 of the drill quickly and repeat as many times as it takes to drll the length. If you can get in a milling machine use a carbide ball end mill they work great with no heat necessary. But remember use the stop or you will just break another one. If you are going to go bigger to a liner size than just get a carbide ball end mill and just drill it out like it was not even there. Try to use a carbide ball end mill.