How do I get iCloud Two-Factor Authentication to work? (cannot turn on to begin using)
You need to go to www.appleid.com (or www.myappleid.com) and sign in to your account. If you can get in, there you'll be able to enable or disable Two Factor Authentication. If you want it on, turn it off then enable it again (if it was on). 

Make sure you delete any devices you no longer have while you are on that web site (near the bottom). It'll make things easier for you, especially with Two Factor Authentication.

If you do this and you find that you are still having issues, call Apple Support and they will investigate your account and see why you are still having issues.