Errors like "Failed to secure existing connection for <site>. Cannot find the requested object." and "Failed to secure existing connection for <site>. The specified network password is not correct." are returned from Windows, which is apparently having a problem accessing the private key for the certificate.
1> Ensure you have all updates from WindowsUpdate installed.
2> Do you have any 3rd party encryption or SmartCard software (e.g. "Entrust") installed? If so, consider removing it to see if it is interfering.
3> Use Tools > Fiddler Options > HTTPS and untick "Decrypt HTTPS traffic." Click the "Remove Interception Certificates" button. Accept the prompts to untrust and remove the certificates. Next, re-enable HTTPS decryption, allow the root to be recreated and retrusted. Then see if the problem goes away.
4> If the problem persists, try installing the Fiddler Certificate Generator plugin. Use the QuickExec box and type:
prefs set fiddler.certmaker.bc.Debug True
and hit Enter. Restart Fiddler. Watch the log tab as you attempt to navigate to a HTTPS site. Are there any new or different error messages?
If problems persist, your last resort should be to attempt to clean up Windows' private keys. A mistake here could cause serious problems for your PC, so make a backup and be very careful, and if you're unsure, DO NOT PROCEED.
After closing Fiddler, open the subfolder named S-<randomstring> within C:\Users\yournamehere\appdata\Roaming\Microsoft\Crypto\RSA\
Look for the recently-created files in that folder; open each in a hex editor. For each such file, confirm that the file has the text FiddlerBCKeys within (indicating that it is used for storing the private key for a Fiddler certificate). Move such files somewhere harmless (e.g. a backup or temp folder) and restart Fiddler. Confirm that Fiddler generates and uses certificates properly again.