Whirring sound especially when air conditioning is turned on

Two possible issues.
Starter relay (what puts power to your starter when you turn the key) or
Your starter is bad.
I would start with the starter relay because it is cheap and easy to replace.
The whirring sound could be your fuel pump or the starter turning but not connecting to the engine to turn it over. Depends on how loud the whirring is...

Good luck.
Semper Fi,
Z

Year/mileage/ service condition etc. The electric pump can make a buzzing / whirring noise but you should barely hear it under the car. Its usually in the tank. If its high mileage or you are just unlucky it can be failing. A failing pump won't overfuel. I think you have more than one system fault and you should get trained help. Modern fuel systems can have anything around 50-60 psi in them. Take care.

Have the power steering pump checked for excessive noise by a shop locally.

In every likelihood have the Bendex of the checked up. It is not connecting the self starter motor with the engine crank shaft thus the whirring sound.

a whirring sound can be a bearing on a component under the hood.(power steering pump,alternator,idler pulley bearing,etc),it can also be a serpentine belt from a worn belt or belt tensioner a cv joint won`t make a whirring sound(it will cause a clunking noise or a vibration).

The A/C compressor robs alot of Horse power from the engine...
Defrost will engage the compressor too...
The whirring may be the coolant fans engaging (drawing air accross the A/C Condenser ).

All of this falls into the ' normal opporation ' catagory , and should not warrant any worries.

It sound like the lobes on the starter's shaft have worn down and aren't making positive contact with the flywheel anymore. A "whirring sound" would be the starter motor spinning but not "catching". If it's a clicky/buzzing sound then it would more likely be the starter solenoid, low battery/bad alternator.
The "whirring" sound should still be winding down for just a second after you turn the key off. The "buzzing" symptom will stop instantly when you turn the key off.

it is not the steering pump because you are not moving the wheel it is the idler pulley it might need replacing check to see if the belt is in good condition if it is just use belt dressing on the pully to see if the sound goes away if it does then the idler pulley needs replacing

check the level of the power steering fluid,check belt tension & condition

sounds like your timing belt has snapped, allowing the engine to tutrn over but the belt is not turning the vales, hence no start