Turn off the gate and do some experimenting to nail down the cause. Move the guitar around... Closer, far, from the monitor, ax8, etc. My guess is the issue is the proximity of the guitar to the computer.
It sounds like you don't have much flexibility in your space, so using the gate and maybe look into some kind of shielding.
The gate will take out the noise, but nothing is free... You will lose the tail end of notes sooner, as they die out. So most of us find a happy medium of what we're willing to put up with, and still gate out most of the noise. Adjust the gate with that in mind.