That being said though, it appears that the green ground wire from the cord does come to the case inside of the kiln (albeit, it looks like a poor connection). Which means that it is probably wired the way that the diagram shows, the 3 prong plug is just not suited for the 4 wire cable. I'd guess the previous owner had a 3 prong receptacle, and opted to match it, and by doing so, cut the ground wire in the cord, and put the 3 prong on to match. It likely worked, but of course, means the kiln isn't grounded.
If the wiring to the kiln is correct (providing a proper neutral, and ground) then a new plug or 4 wire power cable would likely be the solution, assuming the inside is wired correctly.