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When I replaced my mirror glass (after it got smashed by school run mums) I just bought some male and female spade connectors from maplins to patch in an extra piece of wire. I then covered the join in insulation tape, then waterproof gaffa tape, so that it will be easy enough to undo if needed. I also replaced te old spade connectors on the existing wire, as they were heavily corroded.
 
The advantage of doing it this way is that it can be done quickly with a pair of needlenose pliers. No faffing with solder.