Here's a link to "How Things Work". A modern version of the Packard system with plastic instead of brass. In the Packard system, while the principle is the same, the pegs and gear they speak of in the how it works article are a bit different in construction and actually hidden inside the respective wheels.
http://auto.howstuffworks.com/odometer1.htm Since the trip tenths still works, the shaft and drive gears are intact. It sounds as if one of the gear/peg combinations which activates the next wheel in the trip reel is either broken or worn to the point it can't complete the movement.
The mileage and trip reels are almost identical, main difference being the trip reel has the tenths wheel along with a spring loaded input shaft that disconnects from the drive shaft and meshes with the reset shaft when the reset knob is pushed and turned.
Probably the best option is to send to a speedo repair shop and have them repair it.
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