If you have spare 270 Ohm resistors, just use those for the base current. As long as you don't draw too much from the Arduino, you're not going to destroy it by driving it into saturation -- in fact, that's usually a good thing when switching power!
Also, I would use a BS170 transistor (MOSFET) rather than that NPN. That's my personal preference, though. (Btw: the hook-up would actually look exactly the same, as the pin order on that MOSFET is D-G-S)
I would hook two LEDs in series (for 7 V drop) and a smaller resistor (~50 Ohm) to get more light out of the juice in that battery. Three strings of two LEDs in series each is more efficient than six strings of one LED each like you have now.