The keyboard working during the POST process means the BIOS is not having any problems recognizing the USB devices, this is the reason why you are able to use the keyboard in order to access the BIOS and the One Time Boot Menu as mentioned.
The problem you are experiencing seems to be an Operating System corruption since at the hardware level, the devices work.
Check the BIOS under Configuration > On-Board Devices > USB. Be sure the USB ports are Enabled.
Disabled ports will still work during POST.