Just in case anyone ever searches the forums because their NiCad only holds 30 seconds of charge and they cannot charge it past a couple minutes. Drain the NiCad completely.
I guess all of the symptoms I described in my posts above are symptoms of a neglected battery.
Yesterday I rode to work and back (20 km) with full throttle applied. There was no noticeable assist, but I was draining the battery. I was able to charge it for a good couple hours (maybe 3) in the evening. This is the first time in a couple cycles.
This morning I rode to work (10km) and had assist all the way in. No weirdness with the throttle LEDs or lack of power.