The only book I know which is like this is a very popular Young Adult novel titled Island of the Blue Dolphins, by Scott O'Dell.
It's the same as what you're describing (sort of! ), except that the "place" is one of the Channel Islands, off California's Pacific coast, across--and within easy sight of land on a clear day--from what is now Santa Barbara. This was based on the true story of a Chumash Indian teenager who spent several years there, surviving alone, and was eventually rescued. (Much later, after she died, she was buried in a grave which is now considered part of Southern California's history.) What you describe, however, is similar to her actual story (and the plot of this novel).
Except that California's Channel Islands are a whole long way away from the American Great Lakes!!!