make sure you are clicking on the red icon at the top when you run reason 6.51 for the first time because if you just click on the middle blue icon it won't install Authorizer 2.0. I didn't have to reinstall to get it to work, but it did not work for me until i ran CodeMeterHealer.exe.
after i ran CodeMeterHealer.exe I shut down reason and restarted it which brings up the 3 options again so this time i click on the red icon at the top and it took me to the reason website... i logged in (using firefox) & the web auth plugin loaded but you have to click the "reload" link to get it to actually run. then if it works it will have a button that says "Write" (write the new key to your Ignition Key). At that point it should work and Reason 6.51 ought to come up. Then you can close reason and reopen it without the authorize screen coming up as it will just go straight into the program since you successfully updated your Ignition Key.