I happened to have a new heating element resistor lying around, so I decided to do some basic checks with a multimeter. It was showing up 7 Ohms (it's rated 6k8 so close enough). The resistor that was inside the extruder showed as infinity, that's what I'd expect from a resistor that stopped working.
Next up I got my infrared thermometer and decided to do one last check of the existing resistor, then I connected the new resistor to the printer's wiring to see if it would work. The existing resistor showed cold still, the new resistor immediately started heating up - it actually got red hot!
So that confirmed it for me - the old resistor had to be replaced.
When I cut one of the pins and pulled out the resistor what I found really surprised me. It was completely shredded! Side by side with the new one it looks like a saw cut it to pieces.
I decided to connect the new resistor with slightly longer leads. The wires I used were from an old PC Molex connector. It turned out quite nice.
I wasn't sure if these wires could withstand the high temperatures so I insulated them with some Kapton tape.
I added more Kapton tape around the extruder and heated it up for a test. It worked fine! I've had the 3D printer running all day now making quite a large print without any issues so looks like the wiring isn't melting!