Normally, the "whining" or "high pitched" noise comes from the outboard power supply, although it may be difficult for you to tell where the sound is coming from.
Temporarily, disconnect and unplug the outboard power supply "brick" and move it to another room and then plug it in there to see if you hear the high pitched sound from the power supply.
If the sound is from the power supply "brick" that you are checking, it has failed and will need to be replaced. It might be worth the $20 gamble to order a replacement power supply to see that might get things going again. If the seller offers a refund if the device does not do what you want, then it would definitely be worth a try.
http://www.ebay.com/itm/ORIGINAL-OEM-APPLE-AIRPORT-EXTREME-BASE-STATION-POWER-SU PPLY-AC-ADAPTER-A1202-/370810566512
However, if the power supply is quiet when you move it to another room to check it, then the "noise" must be coming from the AirPort Extreme itself. Unfortunately, repairs are not feasible on the AirPort Extreme.....and even they were, the cost would likely be more than a new unit.....so it is time to replace the older AirPort Extreme with a new router.