I agree with Simon's advice re: repairing it yourself, but I would go the unwind and resolder route rather than joining another length of wire if it were mine.

That pickup has clockwise winding so it shouldn't be hard to unwind (anti-clockwise) one turn of wire from the coil since the broken wire is at the coil finish. Of course you still need to work gently, but it's completely doable IMO.
And the coating on the wire should be remove just as per Simon's post.

Additionally, if you want to minimise the amount faffing around with the coil wire and the eyelet (eg: de-soldering lead; multiple looping coil wire through the eyelet, re-attaching the lead, etc) you could just reflow the existing solder and lay in the coil wire. Done.