I would first try to replace the audio board then see if it works. The most you could do after is replace the daughter board cable afterwards. If that doesn't work a motherboard issue has happened.
I check with a pair of our headphones and if it doesnt work then I restart the computer then try again. If it doesnt work then I simply replace the audio jack.
I would first try some headphones myself. If that doesn't work I would restart the chromebook and try again. If it still doesn't work I would replace the audio board.
First i would check and test it out with two different pair of headphones just to make sure it wasn't the headphones. I would then take apart the chromebook and put a new audio board in to. If still not working i would send the CB to Mrs. H
Try new headphones first and if that does not work replace the audio board after restarting the chromebook.
The first thing I would do would try another pair of headphones, Secondly I would make sure the cables were all in properly and if they weren't I would fix them. Thirdly I would replace the cables to see if the cable was bad if that doesn't work I would try replacing the audio board and if none of that worked I would send it to Mrs. H
Just a Media Specialist trying a blog.