I know the black wire goes to the big half cylinder plate.
The problem is that I do not know which color wire is for right and left channel. What I would do is solder the black wire in place and strip one colored wire.
Plug your unfinished headphone jack into your amp and set the volume to low. Then touch your stripped wire to one of the other terminals. You should hear one channel in your headphones play some sort of sound. Make a note about which channel played and what color it was. And then you obviously know the other color is the opposite channel.
Then you just refer back to the pinout for a TRS connector and solder everything up. And don't worry, doing this with a headphone jack isn't going to hurt you or anything else.