Un-weave the braided outside, and gather it all on one side of the cable so it's just like a separate wire in the cable. Twist it if you like so it doesn't sprawl all over the place.
At this point I'd get some heat shrink tubing, but I don't know what your soldering skills are, and sometimes heatshrink can hide problems and be difficult to remove later.
Solder the inner conductor.
Tape it up very well with electrical tape - more than 2 layers over the bare solder joint and any exposed center conductor wire.
Solder the outer braid wires together (since they are gathered into a single wire on each side, it should be just like the center conductor).
Tape up the whole joint so no conductors are showing.
Testing:
Get a cheap multimeter and use the resistance setting to make sure that the wires aren't shorted to each other. Touch the probes to the two conductors on the connector that goes to the laptop - make sure it doesn't read 0 ohms.
Set the meter to the DC voltage setting and plug the power adapter in (but do not connect to the laptop). Touch the probes to the conductors on the connector and make sure it's putting out as much voltage as it says it should (read the label - probably 14-20 DC).
As long as those two things check out, plug it in and away you go.
Keep in mind that while a bad solder joint SHOULDN'T kill the laptop, it's possible it might - even an expensive laptop charger is cheaper than a new laptop. Weigh your options carefully. I'd certainly do this, but I'm quite experienced in this area. You might want to do smaller (less risky) projects until you feel comfortable with your knowledge and skills before taking on a large power supply.
Either way, good luck!
And next time, make sure it's plugged in when little ratty is around. He'll probably stay away, but if not, he'll either learn a lesson or cease to b a problem (probably just learn a lesson - the braid would prevent much current from going very far in his body).
-Adam
