An elderly man had been driving

Have I made any mistakes?

An elderly man had been driving for about ten minutes on the wrong side of the road before he was stopped by the police. He was so befuddled that he was unable to tell them his name and address. After the police managed to contact his family, they told them they were aware that the old man was unfit to drive, but he had refused to hand over his car keys. He had already had some minor accidents and, whenever he took his car, they were afraid that he was going to cause a fatal car crash.

Bassim said:

After the police managed to contact his family, they told them they were aware that the old man was unfit to drive, but he had refused to hand over his car keys.

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The word "they" could easily refer to the police. I suggest that you change that sentence. Also, I would say whenever he took his car out or whenever he drove his car.