An old model engineer magazine gave details of a small furnace. He used K type, the fibre glass covered ones which are cheap and allowed it to burn/melt off.. K is ok up to about 1300C and it's usually the casing that limits it. You might find some enclosed in ceramic that can go to 1300. The ones in a stainless tube are limited to 1250C.
There is plenty of info on the wiki about thermocouple materials.
Actually I would suspect your furnace uses nichrome elements and gets to about 1100C -1200C. It needs to be kanthal a1 to be able to get a furnace to 1300C and more exotic material to go higher – graphite for instance.
K is usually the thermocouple that the controllers are intended to work with. No harm at looking for ones that can be used with a selection of thermocouples but take care that it's not an option when bought. The data sheets can be confusing in that respect as can ebay listings. Some times used controllers with a high temperature thermocouple crop up on ebay in the pottery section. They tend to fetch pretty high prices.
Myfordboy on youtube has done a video of brass casting using a small furnace.
I have the materials about me to cast my own furnace. On my list of things to do.
John
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