I don't have a specific statistic on hand but I can speak based on an interview I conducted with a federal prison warden (the research was about private prisons and whether they should continue to exist). We went off on a tangent and he spoke at length about how they try to rehabilitate prisoners, but as I said, it mostly doesn't work, and many come back to prison anyway. It is also extremely costly and time consuming. However, those who wanted to be rehabilitated were less likely to come back. He had to perform numerous studies and research in his 20+ years as a federal warden as part of his job to the government.
Though, to qualify that, mental illness is something that can be easier to rehabilitate for, since it is a typically more easily identifiable illness, plain and simple. We can notice schizophrenia and borderline personality and the like. But for most crime and criminals, they do crimes out of near necessity -- Don't have much money and you have no education? Sell drugs. Get arrested and released? What do you come back to? Jack shit. Rinse repeat. We need to look at why people commit crimes at all (and its not because they are evil) and start to make solutions there.
Now, I admit, it could be argued that maybe their rehabilitation methods weren't as effective as they could be, but this is a federal prison we're talking about. They get a great budget since they work directly with the federal government (and thus they're not as plagued by issues from state, local, and private prisons).