Help! Magnet Wire broke in Rokuhan Turnout. Fixable?

Like the title said. One of my turnouts died and after some troubleshooting I found that one of the thin magnet wires (the white one) that attach to the electromagnetic coil had broken off, although I have no idea how. Is this fixable? Since it broke off of the top of the coil, I was thinking of soldering it back on as while I can't find the end it broke off of, as long as I get it connected to the coil in the general vicinity of where it should be (currently thinking top outside rim), it should work. I can't break it any more than it already is, so I figure it's worth a try if nobody has any better ideas. Does anybody have a better idea? I tried taking a picture but I lack a macro lens that could show it.

 

While trying to find the broken end of the coil, I broke off the other (black) wire. Looks like I'm going to have to buy a replacement. Thanks for the help, even if I couldn't get it to work. I wish I knew how this happened. It was just sitting on the layout, minding it's own business then boom, it's dead.

 

I repaired on of these a while back. I guessed that the wire just burned through. I did what rray said pretty much. Honestly I was surprised when it worked again and how long it has worked still. That is some fine wire! Another reason to try and fix it again is the new ones are having production issues.
I think that it is important to wait a second between flipping the switches on these. I've had them not work correctly when I flip the switch, realize it's the wrong one, and flip back hurriedly, I know they don't like that. Maybe that could cause a wire burn through too?

 

Well, I got a replacement (and a spare) from Anthony at Z Scale Monster, but I'm still going to try to fix it as you can't have too many spare parts and it'll be an interesting challenge. Hopefully, it'll work even better than before as this was one of the ones that was derailing locos before I fixed that problem.