I have an N router. The cordless phones I have are all the Dect 6.0 models. But, I powered off the cordless phone/base unit and it made no difference. As I originally stated the transceiver is plugged into the designated keyboard/mouse port on the motherboard, which is well within the 10 meter Bluetooth range (actually less than 2 feet away). However, on the second PC that I tried it on (which worked the same way) I connected the transceiver to the front panel USB 2.0 port, which is physically/electronically closer. Microwave was not operating and in a different room in the house. I've tried new batteries and that made no difference. I've un-paired and re-paired the keyboard and that made no difference.
The Logitech that I have came as a package, mouse, keyboard and transceiver. The mouse works OK, the keyboard doesn't - the Keyboard works but can be slow. Either the keyboard is defective or its the Bluetooth.
As a side note, the Logitech support person I dealt with is located in Denmark