Flash Player installer is a universal installer and installs both 32- and 64-bit versions on a 64-bit OS. If the offline installer completed without errors, it's most likely installed correctly. To know for sure I'd need to look at the installation log file.
First, ensure Flash Player is enabled in Firefox: see Enable Flash Player for Firefox
If it's enabled and Firefox still does not detect Flash Player is installed, upload the FlashInstall.log file saved at C:\Windows\System32\Macromed\Flash AND C:\Windows\SysWOW64\Macromed\Flash to cloud.acrobat.com using the instructions at The specified item was not found. or some other file sharing service of your choice, that does not require a log in to access. Post the link to the uploaded files in your reply.
I'm not seeing any errors in the log file. Are you using the 64-bit Firefox version? You can confirm by launching Firefox and then going to Task Manager. If the firefox.exe process has *32 next to it then it's 32-bit, if not it's 64-bit. Note that the Flash Player (32-bit) item in Control Panel (Control Panel\All Control Panel Items> Flash Player (32-bit)) is just the Flash Player Settings Manager, it is not the plugin itself.
Please try the following:
- Go to Uninstall Flash Player for Windows and download the latest uninstaller
- Go to Installation problems | Flash Player | Windows 7 and earlier and download the latest offline installers for all browsers you use.
- Close ALL browsers and applications that may use Flash Player (e.g. instant messaging clients, etc).
- Run the uninstaller
- When it finishes, reboot the system.
- When the system reboots make sure all programs that may use Flash Player are not running
- Run the installer(s) downloaded in step 2
- Reboot the system
- After logging back in, launch Firefox and go to https://helpx.adobe.com/flash-player.html
- Click the Check Now button
- Is Flash Player detected?