I can't tell from the picture where that wire belongs, but anyone with basic refrigeration knowledge would be able to figure it out.   And yes, that whole area looks clean and sparkly!

Giving this a bump so others here will see this. 

We had a refrigeration expert here for many years, but now you have to pull up the monitor top web site to find him.  His name is Travis.  Tell him Ralph sent you, and he should be able to refer you to someone who is familiar with refrigerators of your vintage.  The monitor top site has a forum for "other vintage refrigerators" under the "miscellaneous" heading, so you should be able to get some advice there.  It's free to join.  Here's the link:

Are those two ends of wire what broke? If so, find someone you know that can solder and just have them solder them together and heat shrink it, then start saving up for a complete rewire. I cannot imagine it would be cheap. Wire itself is cheap, but the process is laborious. You could replace the wires yourself, it shouldn't be too bad of a job, just time consuming(henceforth expensive at appliance technician rates of $45-90/Hr).

I think it's the blue wire that completely detached at both ends.

Click on the link to see a thread that will provide some helpful information about which wires do what on your compressor.  If you can see the terminals where the wire connections are, you should be able to figure out where the blue one belongs. 

A rewiring job by a pro could turn out to be rather costly and may not be necessary.

I think our new friend Kris has moved on.  Maybe the MT forums turned out to be a better resource, but it would be nice to know how things went.