OK, dumb question - is this wire a coaxial type of some sort, or is it just a single conductor that is insulated?
If there is only one conductor in there, I would say that it should be super-easy to fix. Whether you want to solder it straight up and then seal it, or if you want to add a small "jumper" to give a bit more length, that is your call.
If it is a coaxial type with a foil type outer conductor, then you will have a harder time. Not impossible, Id probably pull back some more of the insulation, try to turn some of the foil into a pigtail on either end, then solder a conductor to each with regular wire. THe center conductor should still be very straightforward.
I like using brush-on electrical tape for a few initial layers, as it can get into crevices and keep water out, etc. Id wrap the whole assembly in real electrical tape then for further protection. Others might have other, better suggestions for sealing stuff up.
BTW, if it is just conductors straight up that you need to re-connect, likely crimp/solder/shrinkwrap type butt connectors, then coated and wrapped in tape will give a superior connection, bu take more work...
Good luck,
JMH