DISM /StartComponentCleanup fails, but /ScanHealth reports healthy
Solution 1:
The original poster fixed the problem by rebooting - dism /Online /Cleanup-Image /RestoreHealth
is indeed the command that needs to be run for Windows 8 and newer.
There is another tool that can be helpful in this kind of situation: CheckSUR, also known as the System Update Readiness Tool. You can download it for Windows 7 and before from KB947821.
According to this Microsoft blog post, CheckSUR can usually fix corruption in the component store, but when it can't, it's still possible to use it to identify the affected components. Near the end of the CheckSUR log is a list of packages; these entries correspond to Windows updates and KB articles. Get the MSU from https://support.microsoft.com/kb/NUMBER-HERE
and place it in the \Windows\CheckSur\packages
folder, which you may have to create. Re-run CheckSUR after you've done that for each package.
Note that CheckSUR may take a while to do its thing.