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.
Step 3: Open the Power Brick
To open a glued or welded power brick, begin by securing it in a vise or with a clamp. Now, using a sharp utility knife make repeated cuts along the seam until you have cut through the outer plastic shell. It's important to stop cutting as soon as you are through the plastic so as to not damage the electronics. Take your time. Patience pays off big on this step. I typically spend at least 15 minutes carefully cutting through the entire seam. I focus on one side of the brick at a time. Also, I have never had to cut around the low voltage cable or the high voltage cable connector. Just stick to cutting the long, horizontal seam on all 4 sides.
Step 4: Remove the Enclosure
With the top removed, separate the bottom of the plastic enclosure from the power brick.
Set the entire plastic enclosure to the side for later.
You should be left with a bare power brick.
Step 5: Desolder the Low Voltage Cable
If you are particularly driven you can attempt to salvage the strain relief from the piece of low voltage cable we just removed by carefully cutting along its long axis with a utility knife. If you do reclaim it, don't forget to install it on the good low voltage cable before closing the plastic power brick housing.
Step 6: Solder in Low Voltage Cable
Use your wire strippers and strip away 1/2" of the insulation from the inner, insulated conductor. It's likely around 16 AWG wire.
For this low voltage cable there is only 1 insulated conductor. If both of your conductors are insulated strip 1/2" from the second conductor as well. Since our second conductor is un-insulated, we will insulate it using a 1/2" piece of heat shrink tubing or electrical tape.
Finally we can solder the good section of low voltage cable back into its contacts on the power brick. Be sure to connect the correct wire from the cable to the correct pad on the power brick.
Step 7: Assemble the Enclosure
Step 8: Add Strain Relief
Step 9: Test and Use
Use your multimeter to check the voltage of the concentric power plug at the end of the low voltage cable. You should get a reading near the nominal voltage rating of your Power Brick. For this example the nominal rating was 19V. To find your nominal voltage it should be listed in the fine print on the back of your power brick's plastic enclosure.
Congratulations, it's ready to use! You have successfully rescued the world from unnecessary electronic waste, and saved yourself a few bucks along the way!