I think your calculation might be correct, because the loss of hydrogen isn't accurately modeled by Jeans escape.
I remember in my statistical mechanics class we had the same problem as homework, and a lot of people got the answer "it should all be gone by now", but that was because they erroneously used an atomic mass of 1 (instead of 2 for $H_2$ molecules). At least one person used the atomic mass of 2 and got a quite different answer like "most of it should still be here". Then the professor actually emailed the author of the problem and explained, and he replied back something like "Well I'll be darned!".
So if you calculate the Jeans escape rate for $H_2$ molecules and assume nothing else is going on, you get that most of them should still be here. However, it's easy to think of other effects that could make it disappear faster. For one thing, UV light (and to some extent solar wind / cosmic rays) ionizes $H_2$ and dissociates it to $H$ in the upper atmosphere, which justifies using an atomic mass of 1. I think just as important, however, is the fact that the outermost wisps of the atmosphere are constantly being "blown away" by the solar wind.
(Of course, it's also possible that you just made a mistake, but if you get an answer that's sort of in-the-ballpark in order of magnitude, but still not nearly enough to explain the current lack of $H_2$ in the atmosphere, this is probably what's going on.)