I have two ring chimes that have also stopped working. Tried resetting for more than a minute with no result.
It seems lots of people have this problem!
Either there is hardware problem where something is dieing in everyones chimes, or perhaps:
Perhaps Ring pushed a bad update and killed all out chimes!
edit:
I just spent 45 minutes on chat support and they agreed to send me two replacement units! Get in touch with support and mention this thread.