Well that went a lot better than I thought it would!
Most time consuming bit was finding the old wire that I took out - was lurking in the shed.
Turns out when the wire came off it ripped a bit of contact off the screen itself.
Luckily, the metallic contact part is relatively big on the rear screen so I was able to solder it on a little further down from where it originally was.
Started off by melting the old solder off the wire to effectively clean it up and remove the old bit of contact that got ripped off.
I then applied a fair bit of solder to the screen and held the wire in place while the solder hardened.
Couple of pictures (click to expand). Sorry about the quality.
Before:
After:
Also had time to take the high level brake light off and clean the housing - had some dead bugs in it and looked annoying from outside.
I used a boggo standard 30W Antex iron and some standard lead-free solder so best of luck with yours.