Thank you all for the ideas!
I used a multimeter and everything seemed to be working - so my guess is that there is something wrong from the software side.
I have ditched the Bright Pi without finding what was wrong. I got 6 new leds plus a couple of resistors and connected them directly to the 5v on the RPI + an small on/off switch and everything works perfectly (for ~3 EUR
Now I'm wondering why I even decided to go for the Bright Pi... and what's the role of the i2c interface for a simple IR lighting...
Anyway, thanks again for all the suggestions!