Is the installation returning an error, if so, please post a screenshot of the error (The specified item was not found.).
Please provide the FlashInstall.log file located at :\Windows\System32\Macromed\Flash using the instructions here The specified item was not found.. If this is a 64-bit system also provide the FlashInstall.log file from \C:\Windows\SysWow64\Macromed\Flash (both are required from a 64-bit OS).
Thanks for the screenshot. It's very helpful. Essentially, this is an issue with the internet security settings on your machine (Internet Explorer > Internet Options > Security), and not a Flash Player issue. I Googled "internet security settings prevented one or more files from opening' and it returned numerous hits. I'm posting a few for you, however, I'm not endorsing any of them. I recommend you search your self and select one you feel comfortable with.
- https://social.technet.microsoft.com/Forums/windows/en-US/6bd973a1-38b5-4ad2-bcf5-e90be18c c64b/your-internet-security-settings-prevented-one-or-more-files-from-being-opened?forum=w 7itproinstall
- How to fix: These files can’t be opened. Your Internet security settings prevented one or more files from being opened. …
- and there are many more.
The procedures in the last link allowed the Flash Player update to be downloaded and installed.
It's unfortunate that Windows IE views Adobe as an Untrusted Site and requires a user to temporary disable the security settings for the update to proceed.
Thank you for your help.