Unfortunately, in the U.S. at least ASUS does not sell repair parts to end-users, but you might want to tell us your model to see if anybody else has ideas to find the part from other sources?
Otherwise, consider contacting your
local ASUS support through this page if you would like to send it in for out-of-warranty service, or check if there is an authorized service center near you (currently there aren't very many in the U.S. though).
Don't have any experience with USB headphones myself, but this might be another idea if you want to get more opinions and feedback about it (looks like Amazon may even sell a USB to audio jack converter, but again, no idea how well it works)?