MOSFET oscillation with gate resistance
The gate resistor isn't helping anything (that I can see) so get rid of it! 

The circuit should be reasonably stable (without the 10K resistor) so long as your MOSFET source resistor doesn't get too much lower than it is. If you need lower than about 50 ohms then you'll need to add compensation.

Be sure to check stability with a relatively small square wave perturbation in command voltage, with both voltages in the linear range (eg. 20mA to 22mA). 

You don't indicate what kind of bandwidth you are looking for, but usually LM324 and 'very large bandwidth' don't go together. The 10K and gate capacitance form another pole at about 40kHz and, as you've observed, there is Miller capacitance to the extent the drain voltage (V_power - LED forward voltage) changes with changing current.