Open the jaws of the vice so the metal shell of the magsafe just fits between them. Place the magsafe between the jaws with the cable upwards and the white plastic housing resting on the jaws.

Cut off the cable as close to the housing as possible.

Use a cross-head screwdriver to force the metal section of the magsafe out of the plastic housing

Gently prise apart the strain relief metal band around the remains of the cable and put it aside for later.

Cut down one side of the soft plastic inner and peel it away from the wires and circuit board. You are very unlikely to be able to re-use it so don't worry if it tears.

The image shows roughly what you should now have. The braid is soldered to two places (one on each site of the circuit board at opposite ends - middle and right in the picture); the white core is soldered in just one (left in the picture).