Hrm. Several things.
NPNs should be used in high side configuration. see http://www.rason.org/Projects/transwit/transwit.htm
IE your emitter should be connected to GND, and your collector should be connected to your positive rail through your load.
Secondly you're using a 10k resistor with 5 volts at your base (v=ir) means you're putting in .5ma. Your transistor has a gain of between 30 and 300. That datasheet is unclear to me, but you're saying you want like 500ma out and for 10v 500ma IC it says min 30, no max. I would use the max current on your base and see what you can get out of it and then ratchet it down if you have to.
You might damage your micro if you just open it up so do the math on how much you can take out of your avr. Most AVRs can do 40ma per io pin (check your datasheet) so v=ir 5/.04=125ohm resistor. Try that and see if you can do more.
You mean I've been doing the wrong thing for the last 30 odd years?
The text & fig 2 in this link don't match,but nowhere does he say you can't use an NPN transistor with the load in the emitter circuit.
To the OP: Have you tried running the LED circuits direct off the +5 volt line?
If the +5 volt runs out of puff directly connected,it won't work with the transistor circuit.
You should be able to get the transistor circuit to work if you spend a bit of time on it,
Disconnect the 10k resistor from D10,& connect it directly to +5 volts----Does it turn the LED units on,& stay on OK?.
Try two NPNs in Darlington configuration.
This isn't rocket science,but it's a good opportunity to learn.