First of all, the circuit that I'm using is not meant to be used with an AA battery, as it will cook the transistor. So, the idea was to use a resistor-less circuit that would work with these cells, and have a very low draw. As most Joule Thief circuits will not work well with these low current cells.
Second thing is the toroid, which looks like it is made of iron powder, which have very low permeability, and in order to get those to work, you have to fill them up with windings, not like you have there. Best thing is to use a toroid with ferrite core instead. The small toroids that come in the CFLs bulbs work fine.
Third thing is the solder looks like "cold solder", so I would re-solder your connections, to be sure, and be sure to use resin core solder wire or acid flux.
Not all transistors work the same, even of the same kind, or even bought from the same source. So, try another one, I use the Kn2222A, and place a 1k resistor on base rail (center pin), to protect the transistor from it overheating, if you are going to use an AA battery. But, no resistor is needed if using these cells, instead of an AA battery.
Other than that I can't see well enough to suggest something else.
In any case you might start with the suggestions above.
Plengo: I don't know if you've seen Magnetmans video, but I though it was very well made, to your specs, and right up your alley. He's got some nice stuff on his channel. You guys might want to take a look there also. He made some of Plengo's type cells that outputs 150 mA and can light a lot of leds.
BTW: I thought that I'd mention that the grease that I'm using is not stopping the oxidation on the aluminum, at all, so I'll look for the dielectric Permatex grease that you are using, that can be found in auto parts store.
Also BTW: Thanks to all for your support. I'm not giving up, not just yet, I'm had too much fun. And I'm already living in the sticks with the Indians.
Mangetmans video of Plengos cell replication:
Great little cell.MOV - YouTube