A free tool called DiskInternals ZIP Repair was able to recover a corrupted PowerPoint file (*.pptx) that I had on my computer. I learned that pptx files are actually zip files in disguise, and so using a utility to repair a corrupt zip archive could work. At first I thought it was one of those sketchy bloatware programs, but Lifehacker has written an article about them before, so it should be fine.
Using the program was pretty simple. I just opened it and told it which zip file (in my case the pptx file which I renamed to a zip file), and then the program did all the work and recovered everything. It was like magic!