How often academic inventions have already been invented in private companies?
I expect that such a study cannot, in fact, be effectively done.

The reason is that academic re-invention of a closely held secret will be virtually impossible to reliably document except in particular anecdotal high-profile cases, such as RSA.  


Most secret information probably does not become public at all: I suspect that most obsolete trade secrets in companies eventually simply end up in the trash or shredder, because why would anybody bother doing anything else?
For any secret that does become publicly available, there generally will be a long delay before that occurs.  At that point, the relationship between two independently invented decades-old technologies will be difficult to detect by anybody not specifically motivated, since they likely would have very different terminology and specifics.


As such, I would expect that it is impossible to do any general study on the frequency with which academic research is a re-invention of secret work elsewhere.