Separate the earth braiding from the centre conductor wire, and then strip back the centre conductor insulation to expose the bare wire of the centre wire. Now do the same with the long wire you just cut off - this time you have plenty to work with. It is logical what to do from here - you have to rejoin the wires - braid to braid and centre to centre, the absolute necessity being no shorts - they must not touch. each other - for obvious reasons. Ideally you would carefully solder them together and repair with heat shrink tubing, but in an emergency use anything you have around - twist the braid wires together and the centre wire together but keep the two of them apart - ie isolated from each other. Use Scotch or whatever. This is strictly a temporary emergency job, but if you are in the middle of the jungle with nothing but a penknife, surrounded by aliens and must get your Macbook charged no matter what, or you will die, then this could save you. You can buy a new charger or do the proper fix later.
QUICK AND DIRTY VERSION part 1
If you have read this far down, and you need a quick and dirty fix without opening the case - then try this. There are only 2 parts to the charger wire - the centre wire which carries the DC positive voltage, and the surrounding wire braid, which is the DC negative/earth. Touching neither will not hurt you, and all you have to do is rejoin them. The break usually occurs at the point where the wire enters the strain relief bush on the charger - this is where is usually breaks, and is usually visibly obvious. If this is so, then cut it off there. The charger has short circuit detection and protection, so it should usually spring back to life if you repair the damage that has either broken the connection, or shorted it. Now carefully cut away the sleeve of the strain relief bush to expose about 12mm of cable sticking out of the charger - this is the critical bit - you need enough to work with.
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