Keenly is used to describe things that are intense or deeply perceptive. If you're keenly aware that your sister is grouchy in the morning, you'll steer clear of her until lunch time.
Sometimes keenly means "enthusiastically," like when you keenly anticipate the next installment in your favorite science fiction book series. In the U.S., it's more commonly used to describe a profound kind of intensity. A movie character might be keenly desperate; your friend may demonstrate how keenly intelligent she is during her debate; or you may keenly regret telling your grandmother what you really thought of her new tattoo.