Tell him there are over 1,000,000 different fonts to choose from (maybe even more then that). Looking for the font he orginally had is like looking for a needle in a hay barn which has stacks of hay up to the ceiling. I definitly would charge a "exact font find" for this or make something up similiar to that and charge him for your time on this. You probably have 6 hours into searching for this font. If he could supply you with the name of it, it would be a different story, but he didn't and your losing time that you could be working on something else.
I really hope it's just not one sign your making...it better be the whole hospital or something worth the trouble and stress your going through.