Wash everything really thoroughly in hot water, to get any vape residue off (obviously detach from the mod first). You are now in a situation where I can legitimately recommend one of my favorite vaping tools, get a pair of hemostats. They are narrow enough to fit inside tanks, and you can then clamp them firmly on the (remainder) of the coil, and use all the leverage needed to untwist whatever part of the coil is sitting in your tank.
I've accomplished this task often with "stubborn" coil bits, the amount of leverage is almost perfect, and you don't need to worry about sliding "off" the coil, as you've already "clamped" enough pressure.
I'd be really shocked if you were to fail using this method. It has worked for me 100% of the time (and for other things, too, like helping "nudge" or "stretch" an O-ring into the correct place. They are one of my most versatile vaping tools, although I must say, I always wanted to be a doctor. LOL. Harbor freight will sell you a pair for like 2 bucks, and a medical equipment store will also work (they will be pricier.)
Good luck,
Anna