The difficulty you are running into here is an issue of dual addressing. What I mean is, even though you know the gmail account you have been using with the apple TV.. Its not the Gmail accounts password ***** it needs.
Instead, it needs your apple id password, and your apple id is probably your gmail.
So in effect you have two passwords, one for gmail itself. And then another apple id password ***** that gmail account when you use apple products.. Confusing yes?
What you need to do is go to this page
Enter your gmail address as the apple id, and then have APPLE send a password ***** to gmail. ( this wont reset gmails password.. But it WILL reset apples password ***** associates with gmail )
Once you have set a new apple id password, use that new password ***** your apple tv and you should be able to sign in.
Now, I dont intend for you to struggle through what I have wrote.. When you are off work tomorrow Id like to directly work with you every step of the way till we solve this issue.
So I will look forward to hearing from you tomorrow, and you will have my full support in resolving this issue, from beginning to end.
Senior Technician and Network Switching Systems Operator / Maintainer.