Small bench vice
Decide how long the cable needs to be, and cut.
If the heatshrink does not fit over the connector push two pieces onto the cable.
Unravel the braid and twist together. Tin this part
Remove the cotton protection from inner core.
Strip about 5mm of each end. Tin these and trim the earth braid so it's the right length.
The order of the following steps are my preference for soldering mini jacks.
Solder the earth braid to the lug.
Using a multimeter, check for shorts between the pins
Repeat all the above for the other side.
Check for continuity between the two sides. Your meter should show a short from side A to side B