That's actually a plug. No big deal. Tip, should be the left channel. See:
https://en.wikipedia.org/wiki/TRS_connector My instructions will be assuming a repair like your contemplating and my suggested one.
What can happen, and I hope it didn't, is that the jack on the device got damaged. That was typical on the Walkman I had. That repair wasn't fun at all.
You know what I might do instead? Since the wire isn't long enough and your going to have difficulty soldering, you might consider buying this:
https://www.radioshack.com/product/index.jsp?productId=2102949
I would also get some heat shrink tubing for the wire and the cable. The tubing will shrink at least 50%. I don't like Radio Shack's Heat shrink. You may have to resort to
www.Digikey.com.
I suspect that you will not be satisfied with any repair that you make directly to the jack itself.
That said, the wiring probably is one bare with a color. You should identify the channels first. The bares will go together and will attach to the shield or the largest part of your plug.
Stripping wires this fine will not be fun at all. I would reccomend using a warm razor blade to scrape the insulation.
If you can, just hold the shield and then one of the red or green to either connection on the plug to pair the color to the right or left channel. Make sure the plug wiring was damaged and not the jack portion in the player.
If your going to do the wire to wire repair, then cut the 6' cable to the length you need. Put a piece of heat shrink on the cable far away from the splice. The wires will have to be about 1.250 to 1.5" long.
If you have a "real shield" on the cable that you purchased, what you do is open the shield and pull the wires through the small opening that you made in the shield.
Make sure you lay things out that when the cable is layed end to end that there is about 1" of overlapping wire. and about 1.5" of insulation on one side (the side you'll slide the heat shrink on). The other side could be about 3/4 of an inch or less.
The heat shrink will be placed on the cable and on each individual wire away from the joint. Take the wires and cross at the halfway point and twist in opposite directions. Solder. WAIT for the joint to cool. Slide the heat shrink over the joint. Heat the tubing with a match. Do the same for the other connection and finally the shield.
The shield does not need heat shrink. You just need a piece of heat shrink tubing to act as a cable sheath.
If your willing to splurge for a cheap headset, I would cannabalize the wire from that. e.g. **broken link removed**
Re-iterating. I think a splice would be better than replace the end. Make sure it's not the jack (female portion in the player) that's defective.
If the wires look like they have a varnish like coating on them, you cannot solder these without either scraping the varnish off, burning and scraping or using Formvar varnish remover. Trying to solder these would be the same as trying to solder an insulator. It just won't work.