The phrase “how many kilowatts have been used” is meaningless. You probably mean to say “kilowatt hours”, which are an entirely different sort of thing. Trying to measure energy in kilowatts is like trying to measure height in pounds.
Back to the OP, any sort of appliance at all could have high consumption if it’s got a short circuit in it. A direct, solid-connection short will usually blow a fuse, but you could also have a short through some path of low but nonneglible resistance.
If some of the refrigerant has been lost from the system, that could reduce cooling efficiency and the thing could be running constantly without any extra chill on the cold side.
Another possibility is poor ventilation of the hot side - if the fridge is pushed into a small space, completely enclosing the back - so there is nowhere for the warm air to escape from the radiator coils, again, the thing will have to work harder/longer to pump the heat - without necessarily any noticeable change inside the cold part.
Does he have electric heat? In an old house I had a heat pump that was slowly leaking coolant until finally the pressure got too low for it to work properly, so the HVAC was running on ‘emergency heat’ all of the time. The indicator light was on, but since I wasn’t familiar with heat pumps and the weather was cold I thought it was more of a ‘this is what is happening’ light than a ‘this should not be happening, get it fixed’ light and spent several hundred dollars in high power bills before I got it fixed.
Another source of high electric bills would be a water leak in your hot water pipes. The water heater is operating correctly but continuously.
Assuming an electric water heater.
Any updates? Did you find the culprit?