Hmm. Well, I got the Excel Solver routine to fit this curve to your data using a "manual" least-squares fit, and it's terrible. Moreover, the "hyperbola" doesn't even approach the alleged asymptote, nor does it approach the x-axis! The idea is sound, I think, but I have an error in its execution. I've spent too much time on it today already, so I'm afraid it'll have to wait.
Very fun problem, though! Thanks for posting it.