First, the lesson! There's a hinge inside the pop-up flash, and there's a non-moving bushing that holds the flash arm against the camera body. When the arm raises, it rotates a leaf-switch around the hinge axis, and a small peg on the bushing - which doesn't move - hooks the switch and causes it to close a circuit. This signals to the camera "Ho! The flash arm is UP! Activate the flash!" When that small little peg broke off, the switch didn't work and the camera never realized the flash was open.
So I opened it up and shoved a u-shaped piece of paperclicp into the hinge. It's wedged in tight so it doesn't move, and now it hooks the switch in the same way the old peg used to. It can't move farther in or out of the hinge, can't rotate and doesn't pinch any moving parts. It's perfect!
The upshot of all this? I fixed my flash, and saved over three hundred bucks, which is what Nikon wanted to fix the damned thing.