What are my options for removing this broken lug nut?
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Unfortunately (or maybe fortunately, depending on your viewpoint), that is not a broken lug nut, but the nut which is formed into the plastic hub cap. Take the hub cap off and you'll see the lug nuts. If you notice, there are seven lugs on the hub cap. When you take that off, there will be eight lugs on the wheel itself. This is a standard hub cap for the large Ford Econoline vans (like 12 or 15 passenger).
The only way to fix this at this point is replacement.
Here is an image of a replacement hubcap. Notice that it includes decorative lugnuts as part of its design: