How to drive the gate of a MOSFET with a PWM like signal
Summary of solution


The 10 k series resistor will restrict the ability to drive sufficient gate voltage to most MOSFETs - this should be replaced by (say) 10 ohm or 100 ohm)
This will also improve the high frequency activation of the MOSFET due to the low drive impedance charging gate capacitance up a lot quicker.
The LEDs will look dimmer at a high frequency 50:50 drive compared to being switched on at a slow rate due to the persistence of the retina
Using a DVM to measure 6 volts across the LEDs indicates that at a high frequency, the actual wave form is a fast on-off 12 volt signal that of course, averages to 6 volts. Ditto the voltage across the MOSFET.




Picture originally posted by OP: -



For a start the series 10 k resistor that connects to the Nano's output can be reduced to circa 100 ohms. At the moment it halves the drive voltage from the Nano and this will mean poor performance from the MOSFET in terms of switch on resistance.

If the Nano can produce 5 volts logic drive then you should be OK with the IRFZ44N: -



At higher frequencies, gate-source capacitance will degrade the signal seen at the gate if the driving current isn't sufficient. The gate source input capacitance is nearly 1.5 nF and with an effective 5 kohm source, this forms an RC low pass filter of cut-off 21.2 kHz. Significantly higher drive frequencies will turn into a mushy DC level at the gate. Try removing the series 10 kohm resistor and replacing it with 100 ohms for a start.