I'd still like to confirm that it still doesn't do anything if you click inside the window to see if it works then.
If not, try using a different USB port (but make sure you use one on the back side of your PC).
One last thing I could suggest to try, is to close the Logitech G-Key profiler program (assumes the G110 also uses it)
You won't be able to use the G-keys probably, but it just might help.
If that doesn't help, then I'm afraid you'll gonna have to knock at the Logitech forum's door to maybe find a answer.
But in any way, my G19 doesn't have any problems at all (and that basically the same as your G110, but with a added LCD display)