The PWM frequency is not so important, what matters is the rise times of the switch controlling the PWM. Assuming it is some sort of MOSFET, one can assume very fast rise times of 100ns to 10ns or even less.
1 meter of wire has a self inductance of ~1.5µH to ~1.8µH for a reasonable range of wire gauges. This inductance will be some what less if you run the power and ground cables directly next to each other which will cancel some of the magnetic field and thus reduce the inductance.
Unfortunately, LEDs are still diodes and they have some pretty nasty capacitance and non-linear voltage vs current behavior, so that coupled with the likely fast rise times of the PWM switch is going to cause the cable to ring at each turn-off transition with high frequency harmonic content.
It is impossible to say what the fundamental ringing frequency will be, it will depend on the cable and the LEDs, but it will ring at nearly the full LED current then decay over some microseconds, similar to this:
The ringing frequency is probably going to begin at single digit MHz then have harmonic content all the way to the maximum harmonic as dictated by the rise time of the switch. The actual power at harmonics that a 1 meter cable would be an effective radiator at would probably be fairly low but given the current levels, potentially still worrying.
If you don't want to take any chances, or you need to pass FCC (don't cut corners if that is the case, that is an expensive test to fail!), then your best option is to simply ease the PWM edges and slow them down a bit. A bell rings when struck, but distribute the force over more time by hitting it with a pillow and it stays quiet. That's exactly what we want to do here. The LED's capacitance interacting with the cable inductance is the bell, so if we just avoid hitting it too hard, it won't ring.
For what I assume are currents of no more an amp or two, the best option in my opinion is the tried and true RC snubber:
This will dissipate the ringing as heat in the resistor, so it does have a small impact on efficiency. But the snubber pictured will dissipate tens of mW since it only dissipates power when there is a switch event. At 2kHz, this is not very often. It will eliminate the ringing almost entirely:
Not bad for 2 very cheap passive components!
Be sure to position them at the LED side of the 1 meter cable, if you put it on the LED driver side, it won't do any good.