Detroit is not the bad place the national media makes it out to be. Im not familiar with the Corktown Inn, but am very familiar with that area. You should definitely check out Slow's BBQ on Michigan Avenue near Trumbull. You are a short cab ride away from a lot of the attractions
Casinos, Mexican Town, Campus Martius, the Renaissance Center, Greektown, New Center, the Entertainment District near the stadium etc.
As a tourist you should definitely check those places out. All are pretty safe during the day. At nighttime you're best off hanging near Greektown or the Entertainment district. Not that the other areas are that bad, just these areas offer a lot and are very well lit and safe, aside from the occasional pan handler.
Hope you enjoy the D!