The photographs unfortunately don't shed any more light on the situation.
The controllers will likely either be XMTE-2001 or XMTE-2002 - the difference is either a thermocouple or a resistance type temperature input.
As long as both controllers have the same part number you'll be ok.
The pinouts for your new controller are attached:-
So input on terminals 13 & 14
Output on terminals 9 & 10
Temp pickup on terminals 1 & 2
If a resistance type pickup - which I'd guess is most likely, then connections on 1 & 2 are not polarity conscious, if a thermocouple they will be.
The fact that you have no pilot is indicative of a controller failure, though that may have been caused by the element failing!
Probably worth doing a continuity & insulation & short circuit test on the element before risking the new controller...
Edited to add:-
Having looked again at the new photographs with my eyes switched on, I see that it's a XMTE-2301 - which is similar to the XMTE-2001 but with a time proportional output - maybe that means that it's a "learning" controller...
As it also says it's for a type "k" thermocouple temperature pickup.
There is little to be learned from what's left of the label on the old controller, but you can test the temperature probe by measuring its voltage output - about 1mV at 25°C, 4mV at 100°C and 8mV at 200°C - these are nominal figures as for them to be precise the "cold junction" of the thermocouple needs to be at 0°C, which isn't likely to be the case - but they are sufficient to check for a varying output if you should test the element by powering it up before installing the controller (providing that it heats up!).
If there's no voltage output then it's either faulty too (unlikely?), it's not a thermocouple or else there's an issue with the test method/apparatus...