TiboJ:
- Which resistor would be good?
For a TIP121 it is not critical, 1K will do.
TiboJ:
2. The ground wire of the emitter goes to the GND of the power supply, is that a problem?
Not a problem, provided the power supply ground is also connected to Arduino ground.
I suggest you use a multimeter to check the voltage between base and emitter of the TIP121 when the LEDs are supposed to be on. It should be around 1.3V.
What is the voltage of the power supply, and what is the forward voltage drop of the IR LEDs? You have 2 in series, and together with the high saturation voltage of the TIP121, the total voltage you need may be more than 5V.