newshound was thinking very hard :
Yes, good point about wiring in direct. I used plugs and sockets for my
friend because they are in heater supplies switched by temperature
controllers. This way, if either a heater or a controller fails, working
items can be switched over to a spare device.

Your first step, ought to be to check the actual current drawn or check
the rating plate on the back of the kiln. If it is just over the 13
amps, likely it would not blow the plug top fuse, but damage the plug
and socket over time.

One solution for the 'just over 13amps' is to swap the plug and socket
for a 16amp blue plug and socket. These are unfused, so the MCB would
need to be changed if necessary for a 16amp if it is not already a
16amp.