I have had recurring problems updating Flash on Mountain Lion. The ActiveX installer stub fails around the 90% mark.
Based on google results, I see that I'm not alone.
I solved the problem before, but forgot how. So when Firefox disabled the installed version 17.x due to security issues, I struggled to resolve it.
Now that I have, I thought I'd share it here, for those who are having the same problem.
The issue seems to lie with the installer. The workaround is to download the full package installer. It's described as "for System Administrators". Here are links to the latest pkg installer, and the page I found it on. Maybe that page link will contain future updates, I don't know.
The update process is simple. If you've ever installed a full software package on a Mac, it'll look very familiar.
1. Click to open the downloaded .dmg file.
2. Inside, you'll find an Adobe Flash .pkg file. This is the standard Mac installer.
3. Click on this, and follow the install prompts.
You don't even have to stop your browser. Although it may be a good idea, once the install is complete.