Step 1: The Modular Power Supply
Step 2: Breaking the Seal
Starting from the middle of each side, carefully pry at the angle shown while applying some inward pressure (DO NOT HAVE THE PRYING TOOL FACING ANY PART OF YOUR BODY). You may have to dig in a little, but it will eventually pop. Work left to right along the same side, then do the other, and then the narrower sides. Once you have compromised the seal on all four sides, carefully work into the corners until you can start to pry the halves apart. The idea is to try to work the pry tool through the seal, and then on top of the lip to start working the halves apart.
Be very careful not to allow the tool to penetrate too far, or you could ruin the transformer. Also be very careful not to allow the tool to slip and go into your hand or somewhere else.
Step 3: The Guts
Look at the transformer spool (in yellow) to look for overheating/burning, distortion, or other deterioration. This will likely indicate the transformer had overheated. You can test for transformer output by plugging it in on a switched outlet, with only the top half removed. Leave the assembly in the wall-side half so that you can pull it out again without damaging the unit.
No matter what you use any part of the power supply for, never plug the bare transformer into the wall or any fault in the transformer can result in severe shock.
Step 4: Salvaging the Strain Relief
The strain relief is cast onto the cord at the factory, but it not normally glued. The two vinyl-rubbers partially vulcanize during this process, causing the adhesion. You will be breaking this bond, but it's not terribly necessary as you will see in later steps.
If despite your best efforts you cannot separate the cord from the strain relief, you can cut into it with a razor as shown. Try to cut along a spine where you may have the chance to glue it back together again. If at all possible, start from the large end first and you might be able to avoid some extra work.
Step 5: Rethreading the Strain Relief
Do not lube the cable in any way if you can avoid it, as the friction between the relief and cord is somewhat desireable. The structure of the relief makes it act like chinese finger-cuffs, so apply the force as shown to ease moving it down the wire. Again, this will seem tedious, but will be worth the results. Once you get both sides of the wire through, this becomes much easier.
Pull through enough to be able to meet the soldering locations on the board, and then some. You will want to tie a simple but tight knot on the inside and leave a little slack in the case for any slight migration that may occur later. Tie the knot as close as possible to the relief, or slide the relief up to it afterwards, but before reassembly.
If you had cut the relief open and didn't repair it again, or didn't plan to, threading it through is a lot easier, just be sure that the wire is snugly nestled in the form, and not twisted inside it. Once you have properly set the slack you need, use zipties to close the relief, pulled tight with needle-nose to assure a tight fit. You will still need the knot on the inside to keep the wire from slipping and eventually pulling out.
Step 6: Close It Up!
DO NOT USE TAPE, as no matter how good you think your magic tape is, it will eventually fail, especially as the transformer warms up during normal operation. Not even duct-tape, as it is conductive and still won't last. Ghetto-fix is fine, but not THAT ghetto.
To permanently seal it (not recommended), use superglue around the seam, and firmly clamp for at least 6 hours. Once set, fill the groove with a bead of superglue, and allow to dry at least another 6 hours. At this point now the only way to open the case is to break it open, which is why I don't recommend it.
Now instead of tossing that power supply for nothing more than a broken wire, you have saved yourself the cost of finding and buying another one, and have done a small part to not be one of the "throwaway masses".