But what was Elisha's power in the first place? Was God responding to his requests? Obviously Elisha wasn't doing it out of his own power, right?
There is a disagreement about how different nevi'im accomplish different miracles. Sometimes the miracles seem to be either God Himself acting, and the navi only marking the action as or before it occurs. Sometimes the miracles seem to be the navi asking God to do something, which then He does. But sometimes it seems like in giving the navi ruach hakodesh, God has essentially lent the navi a certain extra power, and it is within the navi's discretion to use it as he sees fit to further his mission agenda, so to speak.
My supposition from a traditional context would be that Elisha fit the lattermost category.
Though it's hardly a popular idea from a traditional point of view, it could also be that, in addition to whatever miracles associated with nevuah a navi does, such an individual could be a worker of magic in their own right, as well. Most of the commentators don't approve of magic, and so when it appears in the narrative, they try to read it as miracles. Some-- but not all-- of the Rabbanim felt similarly. But it seems like some of the nevi'im, avot (patriarchs), and other people in the Tanach did know how to accomplish magic of one sort or another-- or so the stories make them out to be. I personally find much more satisfying the idea that the positive things Elisha does are miracles or miraculous deeds done with ruach hakodesh, while these sorts of unjust uses of power represent his own magic at work.