Why does Prospero show mercy to his foes?
It's kind of hard to say, since if you were to line up Prospero's strengths on the right side of a scale and the weakness on the left, it would be akin to weighing a feather and a boulder.
Bad stuff: makes his daughter think she causes the storm that may have taken innocent lives, makes her question her paternity, ("She [Miranda's mother] said thou wast my daughter." I.2.57.). He's a big whiner, "Me, poor man, my library /was dukedom large enough." (I.2.109-110) and mean to Ariel.
Good stuff: Well, he kind of feels badly in the end.
The only reason for mercy would be the kind of forgiveness he hopes to receive.
I have been studying the tempest for a certain amount of time, and prospero shows mercy towards his foes because during the play he recieves a change of heart. He does have some help from his sprite Ariel who basically says "IF you saw them crying now, your heart woul soften" and then says "Mine would if I were human"