Windows 10 stops responding to HID input devices

Bear with me, because I've done some of my own troubleshooting to no avail.

My workstation runs Windows 1607.14393 with all current updates. The machine is a Dell Precision T5600 desktop with dual E5-2620 CPUs and 48GB of RAM.

The symptom: After a random period of time (a few days on average), either the keyboard or the mouse on the system will stop responding. The machine will not accept any input from the affected device. It can be any HID device and it is only one device at a time – I have deliberately left two keyboards and two mice connected and when the problem occurs, one of the four devices stops working but the other three remain active.

When the device stops responding, unplugging and replugging any HID device also causes it to stop responding. For example, if I have two of each device connected and one mouse stops responding, disconnecting the other mouse and reconnecting it causes that mouse to stop responding. No hardware insert or disconnect sounds are played.

In Device Manager, when this occurs, there are several HID "consumer control devices" with exclamation points on them. All of them have an error message reading "No drivers are installed for this device." When looking at the USB VID/PID of the devices with errors, they are always those of the device(s) which have either stopped responding or were disconnected and reconnected. Obviously when the HID stack is functioning, these same devices appear in Device Manager but do not have any errors associated with them.

I tried to uninstall the devices thinking perhaps this would kick Windows into resetting the devices after a disconnect/reconnect. However, the "removing device" dialog stays on screen indefinitely and never concludes. I have waited over a half hour.

In either case, as soon as the HID devices start misbehaving, it is no longer possible to reboot the machine. The machine will hang at the Restarting screen with the spinning circles. I have left it here for over an hour.

No events whatsoever appear in any of the event logs. I have accessed the logs while a device had failed, and also after a reboot, and saw nothing at all appearing.

The only remediation I have is to force-reboot the machine by physically powering it off.

This has been going on for about 2 weeks now. The problem even caused a hang on the restarting screen due to an automatic update happening while I was away from the machine. The update did seem to finish successfully after I hard-rebooted the machine.

Does anyone have any advice on further diagnostic steps I can take to figure out the problem? Reinstalling the OS would be extremely tedious at this point given that I have some licensed applications which would require me to contact vendors and "explain myself", as well as a lot of configuration customizations which would take a long time to repeat.