>>I assume that the gear is broken in the wiper motor.  Crap.

>>The motor runs fine, but the wipers do not engage.

Blake,
Check the wiper arms where they meet the drive shafts the stick out of the front of the van.  I have had mine come loose and not enbage the splines on the drive shafts and flop around as you describe.  Take the arms off clean up the splines and reattach firmly.
Way easier if it works.
George
'85 Westfalia