Well for one thing, most things are "on" these days once you plug them in. Unless there is a hefty switch on the back you flip, then it is probably powered up enough to detect you pressing the "on" button. My guess is it uses a switching power supply that is generating enough noise to swamp the RFID radio signals.
I recently purchased an HP touchscreen PC, so I tested my ID-12 reader both near it and further away. In my case, the reader fires off at about an inch from the token pretty consistenly. And if I placed the reader right in front of the screen, either in sleep mode or powered on, it still behaves the same. However, my Arduino was powered by a battery so I could walk around with it.
The idea of the radio-suppression chokes clamped around the power wires may well help (or the USB wires for that matter). Any stray RF shooting along the power circuit might be damped down enough to make it work.