Damage typically occurs where the lead exits the power adapter housing around the plastic strain relief caused by repeated stress from acute bending or pulling of the cable. The outer plastic insulation breaks exposing the braided wire strands of the ground (-) wire and very often these strands fray and break resulting in disruption of current flow.
The inner live (+) wire is often broken inside the insulating sheath indicated by a knot like kink.
What you need:
Mallet or hammer
Plastic pry tool with chisel shaped end
Electrical insulating tape
On HP and Dell adapters there is a narrow channel between the housings where the blade
can sit so limiting collateral damage to the casing surface.
After repeated tapping, varying the amount of force as necessary the casing halves wiil begin to separate. Repeat this process on the opposte long side of the housing.
The casing ends can be separated by tapping the plastic pry tool into the corner gaps
made previously. The tool itself can be made from any unbreakable plastic about 2mm thick (in this example coathanger plastic).
as much as possible of the positve and negative leads connected to the circuit board. Cut
the wires to equal lengths and strip away 3mm of insulation. The braided outer ground
wires should be twisted to form a single wire. You can also tin the wire ends with solder
if a soldering iron is available. Any remnants of the strain relief should be trimmed off if it cannot be reused (the usual scenario).
Here the protruding +/- leads were a little too short to be able to do an effective twisted join with the remainder of the output cable. So the solution was to use a small household electrical connector to join the wires which uses screws to lock the wire ends.
Join the wires white to white and black to braided ground wire.
Optionally you can add extra insulation/protection by slipping on small pieces of scrap
insulating sheath from household cable.
shells. Without the strain relief there's a small gap round the protruding leads, this can be filled in using kitchen/bathroom silicon sealant if you have some available.
Finally tightly wrap electrical insulating tape round the repaired area.
One last thing, remind yourself to not treat power cables like string!