Nice one - as a long time member of this exclusive club will have to hone my soldering skills and give it a try.
Good man! I don't know how much knowledge you have on this subject, I had none and these are the main things I learnt:
If you do try it, you can test that you have the right connections by simply connecting the two points shown with a wire pushed in (as long as it makes a contact) and then trying it on the bike (without the dash casing).
I used 'AWG 30' solid core wire which you can buy off ebay (or I can post you a bit, I bought far too much). This was recommended by the electronics guys.
The solder should contain 'Rosin'. Whatever that is.
You will need a magnifying glass, preferably one on a stand to keep your hands free. If you have something to hold the dash while you work it, even better. I didn't.
You will need good lighting.
The soldering iron will need a fine tip on it, I filed mine down to make it sharp but not to a point (a bit like a chisel). Make sure it's clean and tinned.
Number 1 lesson for me - get the soldering iron properly hot, mine took ages to get hot enough. I actually gave up hope off being able to solder a good joint and went to watch TV for a half an hour, but I forgot to turn off the Soldering Iron, when I came back it was hot enough!
Number 2 lesson - make sure the soldering iron can actually touch the copper ring on the circuit board, otherwise the solder will not flow into the joint. I practised about 50 joints on an old PCB from a computer before going to the real thing and I am very pleased I did! 49 of those joint were rubbish or non=existant.
The new wire could break if it is left to flap about on the bike, mine is stuck down with a bit of superglue.
I am told that there should be less bare wire by the new solder joint than I left.
I also scraped the corrosion off the board and left it alone at that.
There's a good soldering video by Manchester University on Youtube, search 'Basic Soldering Technique'. The most useful bit was 'apply heat for 1 second, apply solder for 1 second, leave heat for 1 second - remove heat'.
Good luck. Drop me a PM if you want me to post a bit of wire to you. It's about £3-4 off Amazon.