Step 1: Tools Required
1. Vise (or other means for securely holding your power "brick")
2. Soldering Iron (and solder)
3. Multimeter (not strictly needed, but its always better to check your work)
4. Flathead Screwdriver
5. Wire Cutter/Stripper
6. Utility Knife
7. Hot Glue Gun
8. Super Glue
9. Heat Shrink Tubing or Electrical Tape (not pictured)
Step 2: Get Started!
Now for the disclaimer....
Warning! Throughout the remainder of this repair we will be exposed to circuitry that poses a shock risk. DO NOT touch components or their leads unless you are absolutely sure you understand what you are handling. This is a simple repair that does NOT require even a moderate electronics knowledge you only need to be able to solder. All the risk is in touching components/contacts incidentally or intentionally that you are not instructed to touch. Never try to repair a power supply that is in service or connected to any source (i.e. a laptop or wall power). Proceed at your own risk.
Lets continue by unplugging the high voltage cable from the power brick (if your power supply allows for this), and by making a single cut through the low voltage cable near the fraying cord. Ensure the fraying portion of the low voltage cable is left attached to the power brick so we can remove it.
Set the high & low voltage cables to the side. We will need them later.