I wish I'd done some troubleshooting before I bought a new jack. I got this far in the manual and discovered the connector holding the second ribbon in place had popped open and the ribbon was loose. I reconnected it and problem solved!

If it pops open it looks as if the connector is gone but it isn't. I put the ribbon back in place and gently swept the tool over the spot where the connector is and it went back into place.

Oh well, I have an extra jack if I should ever need it in the future. I know others with the same kind of iPod so I might save the day for someone someday.