Not a problem, provided the power supply ground is also connected to Arduino ground.
The power supply isn't connected to the arduino gnd at this moment.
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.
Voltage between base and emitter is 0 volts, and between collector and emittor 10.6v.
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.
Power supply is a 3s lipo(at this moment 12.4v) and the forward voltage of the IR leds is 1.5v.
edit: the 1k resistor did not make it work.