I have no doubt that you would be surprised. Ron Gardenhire and Terry Ryan both talked about it in the media quite a bit, though. The story is, they decided with about 10 days left in ST, but only told Joe at the end of ST because they didn't want to mess with his head.
Obviously news reports this old are hard to find, but if you are truly curious, go for it.
After a guy has good spring stats, it can be natural for team officials to give them credit for it. But it doesn't really tell us what kind of factor those stats played in their decision -- especially when that decision was mostly a foregone conclusion anyway.
Here are some news reports from the beginning of that spring:
March 3, 2004, the day before the first spring training game, Jim Souhan referred to him as "Joe Nathan, the Twins' new closer."
March 4, 2004, previewing the first spring training game, LaVelle E. Neal III wrote "Joe Nathan, expected to be the closer"
March 6, 2004, Patrick Reusse wrote "Plan A is Joe Nathan as the closer and J.C. Romero setting up."
There was no need to make it official or even commit to the decision privately until the end of spring training, since there's no such thing as a "closer" during spring training games and especially since Nathan was new to the organization and new to closing -- but that doesn't mean Joe Nathan needed that particular spring training stat line to get the job.