WinRAR provides a built-in repair tool for damaged ZIP files. Open the file, then go to Tools > Repair archive.
DiskInternals Zip Repair - Free to download, it managed to recover eight out of the nine original files. It's dedicated to repairing zip files so it has less clutter to navigate than WinRAR and it launches with a wizard-style prompt instead of you having to go through any menus.
Zip2Fix - Tied for first place recommendation. Its spartan interface has even less clicks involved than DiskInternals: you open the corrupt ZIP and the software automatically begins its process and outputs any recovered files to the source directory without asking or prompting about anything.