Just to make sure... Did you check whether the LED can be turned off in the monitor settings?

If that is not the case, I would suggest getting a smart plug, and specifically one that can monitor the power consumption. That way you can automate it in such a way that it switches the power off when the power consumption drops below a certain value and stays there for a bit. I set this up in multiple locations in the house and it helps save on standby power usage. I use node-red to monitor and control the plugs, but other options like Home Assistant would work as well.

The plugs (Shelly) I use even have the option to set up the automation in the app (which requires cloud connection to be enabled), or run scripts on the devices themselves (no cloud required, but not all their devices support local scripts. You typically need the newer "plus" devices for that. You can then easily set up an automation that switches the plug off if it is powered on and the power consumption stays below 10W for 5 minutes for example. You could put your whole desk on that plug, which should save you some money on your energy bill as an added bonus.