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If the Kindle hasn't been updated, or hasn't been onine, for a while, then its software might have become outdated, perhaps. Updating, and/or leaving it online, connected to WiFi, for a while, might cure that.
Also: check the date and time settings. Wrong date settings can make the internet think that you are not authorized, for rather abstruse reasons... correcting the date setting might get the logging in to GMail working again.
If the above doesn't get it working, then also check your password, as to whether your GMail is actually still logged in, and also, as a last thing to try, log out of GMail, then close the browser or email app, wait a few minutes, then open the browser or email app and log in again, then try to send email... that might clear up whatever is wrong.
If the above doesn't get it working and the error is persistent through all those efforts, please tell me more - by Reply To Expert - about whether you are using a web browser app (e.g. Silk) or an email app (e.g. Outlook) and I will advise you further toward a solution.