Update 7/12/18 - After a few day/night cycles the ni-cad is starting to stabilize. The first cycles had the battery discharging all night due to it being fully charged but this morning the led was out yet with just the first hour of sunlight, the battery charged enough to get the led lighting again when I covered the solar cell. So... tonight I will see how long the battery will discharge. I will be content if it will work till past my usual bed time of 9 or 10pm. I will be very happy if it will last past 2am when I usually wake up for work :-)
The bear is alive.....sort of.
This is the circuit being tested except that the solar cell voltage is about 2 volts and the diode used is a 1n914. The Joule Thief is connected where the led is indicated and everything seems to be working well.
Once I am sure everything is good after a few day/night cycles then I will reduce the circuit to a size that will fit inside the bear. The charging/switching circuit in the base and the Joule Thief in the lamp housing.