***** NOT A TEACHER *****
Here [this] is the usual explanation when using the verb "see":
1. I saw her cross the street.= I saw her actually get from one side of the street to the other. I know that she got safely across.
2. I saw her crossing the street. = I saw her in the street. But I do not know whether she actually finished crossing. (My telephone rang, so I turned my head away from the window. Who knows? Maybe she decided to turn back, or maybe a car hit her, etc.)
That explanation does NOT work for the verb "let." Therefore, I am 99.99% sure that native speakers would NOT say "They let her cooling her heels in the office."
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