I can't understand why you're attempting to wire on a second hand piece of cable!
Maplins will sell you replacement jack plugs for a few pence!
Stereo Headphones are wired as follows:
TIP = LEFT
RING = RIGHT
SHAFT = SHIELD
Stereo microphones are wired the same way...
The 'shield', 'screen' or 'ground' wire on a signal lead is effectively earth. Any stray conductance between a signal path and earth will 'deck' that signal....
Mono jack plugs are wired
TIP = SIGNAL
SHAFT=SHIELD
Somebody mentioned that the mic plug SHOULD be mono.... And this would explain why your twisting the red/white wires in your 'replacement' plug doesn't work! EFFECTIVELY what you're doing is decking the signal! On a mono plug the area where the ring is would be part of the shaft!
Plug your'replacement' cable into the sound card, fire it up so you can monitor the mic input on the headphones and (NOT touching the screen with your other hand) touch the red and then the white wires.... touching ONE of them will cause you to hear a 'Buzz'. THAT wire is the signal input....
Strip the other one back and deck it by wisting it together with the screen. Wire the output of the mic to the wire you got the buzz off....
Another way of doing this would be to use the continuity function on a simple meter to establish which wire is connected to the tip..
But, seriously, replacement plugs shouldn't come to much more than a quid! Trim the wire back a fair bit too; if you ripped the original plugs off its possible you've broken the cable further up...