I really liked unkchuk's map idea. It probably won't take much though, since a hospital should be more receptive to something health-oriented than your average workplace. At least one would hope so.

If you have a specific area in mind that seems perfect for the bike repository, you might want to mention it, that way it's one less thing they would have to worry about. Although they may want to pick the spot themselves of course.

Good luck and be sure to let us know how it goes.