I can't see it being too hard: Take something like this:
http://41j.com/blog/wp-content/uploads/2016/08/fg100.jpg
You can get mineature plugs and sockets for use with K thermocouples. They will be yellow.
PCB feedthoughs will likely work because they are at the same temperature. You can start with a bare wire thermocouple and you should be able to make the twists.
Use another thin guage wire, and some way of pulling the made TC and fastening it. Titanium wire
**broken link removed** would be a good choice since it isn't a good conductor of heat.
You could likely find different bead colors or lengths so the polarity is right.
Useful info: A short with any wire on your readout should read room temp as long as the thermometer is capable. Polarity test, attach TC and place between fingers. Temp should go up.
Try not to solder, you likely can't anyway K type thermocouple wire. The junctions (1/2 junctions) should ideally be at the same temperature.
Making your own thermocouple can be done with a high current power supply (car battery?) and a carbon block. Twist tightly and run junction along carbon block. No twists would exist on the final product which would have a bead at the end.
There is something called extension wire. This is not the same as thermocouple wire, but it has the same temperature characteristics.