Getting partial flow; downstream from transistor doesn't seem to be working
Your problem is very simple: you need to provide a higher LED voltage or go to something other than a Darlington transistor.

Look at the data sheet for the TIP102. Vce(sat), the collector emitter voltage when fully on, can be as much as 2 volts at 6 mA. The forward operating voltage for white LEDs is nominally in the 3 - 3.5 volt range. This means that even without limiting resistors at all there is unlikely to be much current through the LEDs, since with a 5 volt supply there is simply not enough voltage available.

Your best bet is to replace the TIP102 with logic-level NMOS FET. With only 5 volts available for gate drive, "regular" FETs may work, or they may not. Try something like an IRZL44N. They are a buck or two from places like Digikey and Mouser.

Alternatively, drive the LEDs directly from the battery, although if you do you should increase the LED resistors to reflect a resistor voltage of 3 to 4 volts.