The cut-away cap idea sound like a winner. you can align the new wire right into the little indentation where the broken off wire is. Put one drop of liquid silver solder there and then heat up the male pin on the cap, let the heat travel through the new wire to the joint with the tube, put a drop of solder on the male pin and watch the whole thing come together.. That silver solder makes a really strong bond.
http://www.amazon.com/gp/product/B00...G964GE853TH8GB
Keep the temperature to a minimum as I believe that high solder temps break down the bond between the glass and the wire as it goes into the tube, as the metal heats it expands and as it cools it contracts. There's your leak.. at least that's what happened to me. Grinding the glass would be the last choice for me.
silver solder is considered a hard solder melting temp is around 840F. Soft solder is below that temp. The seller says that his paste melts at 430F..