This issue appears to be affecting users who upgrade from iOS 7 to iOS 8, however, it doesn't appear to affect ALL users. This leads me to believe that there is likely some sort of update error that occurred during the installation of your device that is causing some software interactions to prevent the successful mirroring of your iOS displays to your TV using an AV Adapter that previously worked.
Therefore, the suggestion I wish to make is to complete a 3 step process of:
1) How to back up your devices using iCloud or iTunes - Apple Support
2) Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support
3) Restore your device from an iCloud or iTunes backup - Apple Support
Keep in mind, Step 2 is not just using the menu on your iOS device to "erase all content and settings." Although that puts your phone back to factory settings, we want to do more than that. If we use iTunes to complete the restore, then not only is the user data cleared, but the entire operating system software for the device. This get's downloaded by iTunes on the PC or Mac, and then a fresh install is completed on the device.
If this doesn't resolve the issue, and you purchased a Apple branded AV Adapter, then I would contact Apple Support of more assistance, informing them that you have completed the steps above.