Read the legal guidelines on this website and be sure to make yourself aware of the risks of this repair. Warning, this repair requires soldering skills and a basic understanding of electronic circuits. Place your itouch on a clean flat and soft surface, such as a rubber mat or a piece of soft cloth and get the required opening tools.
Follow the steps of the guide on how to open an iPod touch 1G until the backplate is removed. Plug a headphone into the headphone jack module. If you have a multimeter and a separate headphone plug as shown in the image you can use the diode tester to find out which of the channels is not making contact. If the audio is distorted, is likely to be ground channel. On this particular repair, ground is being repaired. Which is the first point from the button.
Remove the part of the ribbon cable covering the first (ground) contact. This is a destructive step on the itouch headphone jack module, but you can replace the ribbon cable with a wire once the repair is done.
Remove the first gold pin by using a dentist tool or a strong needle as shown in image. Lifting upward, it should come out.
This is what the gold contact looks like once it's removed from the jack.
Use the dentist tool or a sharp flathead to widen the contact spring.
It should look more or less like this before you reinstall it.
Reinsert the contact spring into the jack module and solder a jumper from the contact to the logic board as shown in image (black wire).
Use a either a multimeter to test or some headphone while playing music to verify if the repair was successful.