Headphones were tugging or yanked enough to wear out the connection, and I found these instructions invaluable. It was enough for me to get the back panel off.
The back panel has the headphone jack (visible in your last image). I stretched the spring connectors out to give more pressure, but the real problem were the pads on the flex connector -- a couple of them had holes worn in them. I applied a light coat (a dab, followed by a touch from the desoldering braid) of solder to the pads, reassembled the unit, and it works like a charm. I hope this helps anybody who has a flaky headphone connection.