As others have mentioned, you need to keep the filter on at night. While you probably won't kill many bacteria, you are definitely not helping the ammonia levels in the tank. And without any circulation in the tank you can also run into oxygen depleted dead spots and temperature gradients. You could try some sound deadening insulation around the housing of the filter which would help, but you apparently have VERY sensitive hearing to be annoyed by the buzzing of the impeller. Home Depot sells Frost King self-adhesive insulation. Looks like aluminum foil but one side is sticky with a dense foam material. I used it to deaden the interior of my Trans Am a couple years ago and it works nicely.
I used to be like you and had a horrible time sleeping through the night. Every creak, rustle, etc. would wake me up and it took forever to get to sleep at night. Then I found the wonderful world of noise machines. We use a white noise (almost sounds like a continuous waterfall but there is no change in the sound) and it helps to block out everything. I would highly recommend you give one a shot, because if it works, you can drown out anything and sleep well.
And there's always earplugs!
