I had a short in my SNES AC adapter right where the strain relief was, so I couldn't really cut out the offending section of cable and splice the good ends together. I'd have to open the housing, desolder the old cable and cut off the offending part, then strip and solder the cable back into place.
In their infinite wisdom, Nintendo used a screw that has a rectangular extrusion with rounded edges. Chances are that nothing in your toolbox will ever fit it, and it isn't remotely similar to a gamebit or Torx bit or anything else. Plus the screws are deeply set into the plastic housing which limits what kind of tools will reach down there.
I used a wide flat blade screwdriver from a Swiss army knife to enlarge the diameter of the two screw holes, which allowed a pair of bent needlenose pliers to just barely fit. After much cursing the screws were eventually removed. It's an ugly solution, but the power supply had been dead for years and it was about time I fixed it or threw it away.
A 12 AWG wire stripper can remove the outer insulation safely without nicking the interior conductor and secondary insulation. When you solder the new cable in place, remember to put the heat shrink tubing segment back on and check the polarity -- the green center wire is negative and the braided copper wire is positive. If you had to remove the strain relief, you can tie a knot that goes inside the housing and prevents the cable from being yanked out.
I'd recommend double checking the output before plugging it into your SNES. It turns out seeing 13V is normal because the 10V power supply has very noisy output which fools a multimeter -- just one (large) capacitor is used for smoothing the rectified DC output. If you are feeling adventurous there are two unused locations for additional nonpolarized capacitors. I'd imagine the 0.1uF ones would be a good match, but haven't had the spares to try them out.
And yes, the blades on the power supply are supposed to wiggle. They are designed to have a range of independent movement, though it initially feels like they are loose due to a broken solder joint. Don't worry.