Can I solder back together a single broken copper winding wire in an electric washing machine motor?
The big issue here is safety- the motor has mains voltage on it. If you can get some clearance around the area and apply some insulation below and beside the break (I suggest Kapton/Polyimide tape) then you can scrape the insulation off the broken ends (may not be easy) and mend the break (without destroying the Kapton/Polyimide) possibly by adding a short jumper wire rather than trying to make a butt joint.



You would then need more insulation over the top (I'd then slather it in epoxy after a quick test), and something like fish paper (rugged insulating material) to provide protection against any possible dangerous short to ground from chafing. 



Personally, I'd give it a try to save $160 and shipping cost/time, but I can't judge your abilities so the safe suggestion is to go buy a new motor.