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I just looked through my archives and found no match. The most unusual earmark of the font is the diagonal stroke of the "N" being raised from the baseline. This is a feature of Benguiat and Korinna ... but not much else.
If you can post a direct scan from a printed image rather than a photograph, then I can run an automated comparison against my FontExpert database. Other than that, I think the client is being unreasonable to insist on a font which he does not know the name of nor anything else about it.
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.
Are the signs engraved or vinyl? If engraved, it could be an engraving font, which I think use very specific fonts or when using common ttf/ps fonts, the serifs for instance become distorted and change the look of the original font. I'm not familiar with engraving equipment, but I thought that these sometimes use unique fonts.