Ditch Light Repair

I doubt Kapton tape will make a connection, but without seeing exactly what you're dealing with, it's hard to suggest a potential repair procedure.

Sorry for the extended delay, finally got time to sit down, peel the thing open and take a look at the damage. From what I can see I would only need to shed a bit of insulation and then just heat the pad up with the iron (that I still need to be and definitely haven't procrastinated buying) and stick the wire back on

Here's a photo of it:

Looks like an easy repair for the LED to light, I can't comment about fixing the physical damage to the light structure.