I spent about 20 minutes uninstalling and reinstalling Mail accounts and carefully typing in passwords only to have GMail notify me that my password was incorrect.
Fortunately a Google search brought me to a forum post telling me about this site.
If you click on the link above, you will be taken to a page that will allow you to access your account within the next ten minutes. If your password is correct (as I knew mine was), it will allow access and, I suspect, identify the new device.
Just so you don't pull your hair out like I did mine, I'm posting this for you.