What is this connector?
The connector is not something special, it's seen a lot in computers, and if you go to your local electronics store1, they probably have the male part. It's called a '2 Position 0.1 Inch Straight Friction Locking Header'. That is, it's a header, with...


2 Position -> there are two pins
0.1 Inch -> the distance between the pins
Straight -> the pins are straight (that is, not bend)
Friction lock -> the header as a locking system which prevents it from popping off


If I'm not mistaken, here is the male part.
(But I'm absolutely not sure! I would recommend you to buy this part in a local shop.)

I don't think there's a this-connector-to-jackplug part, so you will still have to solder a bit. However, if you use the male part, your connector is still intact, and you can still use it in your shirt as well.

By the way, see here for a similar project.

1: I don't mean a hardware store, but an electronics parts store. If you don't have one nearby, you can also try in a shop dealing with computer components, and maybe they have something. If you're lucky, they have an old cable they wanted to throw away and you can get it for free!