Thank you for the details of the issue you are experiencing with updating iTunes. You may want try downloading the latest iTunes update directly from the Apple site. When you are installing it, be sure you are logged in as an administrator:
If you continue to see the same error, I recommend the steps in the following article:
Thank you for using Apple Support Communities.
it goes through all the steps until the final one, then I get an error message "Verify that you have sufficient privileges to start system services",
Try the following user tip: