The red and green wires in this photo feed into the motor. Wiggling the green wire does the trick.
Lisa decided to remove the fan motor to see if she could repair it. This first required removing the windshield-washer fluid reservoir and the engine coolant overflow tank, both of which are visible in the foreground of this picture.
Even after finally wrangling all five holding screws out (a tricky proposition, as it happens), the motor still wouldn't come completely out. There's not quite enough clearance, and what looks a bit like hoses coming out of the light-colored fittings on the right are actually metal pipes for the air conditioner. They can't be removed without venting the coolant, which would be bad (and expensive). Lisa was obliged to remove the outer housing without being able to get the entire fan assembly completely out of the engine compartment. As she pulled it loose, she started to curse as a spring popped out.
Lisa examined the innards of the motor. She said that it appears that the fixture that holds one of the two brushes against the armature of the motor was broken. The short answer is that I really need a new fan motor. This motor isn't old, either; it's what they installed when they put in a new AC system a year ago.
I couldn't get the camera to focus well here. (Lisa later reminded me that for shots this close, you have to disable the auto-focus and set the focal length manually.) However, you should be able to see the spring at left and the block on the right that is the out-of-place brush.
Lisa was perplexed here. Putting the motor back together again was not going to be easy, because the brush simply wouldn't stay where it belongs during re-assembly. She thought about it for a while and got an idea. She got a drinking straw and removed the cover. Flattening out the cover, she threaded it through the motor and used it to pull the brush and its spring into the place they belong. She then carefully pushed the pieces of the motor back together and slowly pulled the straw cover out. This worked and the pieces all slid back into place, but the real test would be to see if the motor would run again.
After re-installing the fan assembly, Lisa told me to try and run it. I turned on the key and then the fan switch: It worked! Amazingly, it had all come back together.
Although Lisa's work has restored the fan for now, there's no telling when it will quit again, or even if still wiggling the green wire will restart it, but at least when it does quit again, I'll know exactly what's wrong. I'll still probably hire someone to replace it because they're apt to have better tools and a better idea of how to get that fan in and out without having to pull the entire air conditioning unit. (I hope.) But we are very unimpressed with the dealership's diagnostic skills. If Lisa found the problem so quickly, they should have found it right away, given that (at least in retrospect), Lisa was trying some of the most obvious things by simply poking at the motor.