On Mac OS, you can try Disk Utility (Applications -> Utilities -> Disk Utility).
I don't have a kindle, but if you are lucky, it appears in the left sidebar and you can remount it again. (My kobo disappears completely).
You can also try the App
Mountain (also on the
App Store, it's $5.99), it works as a menu bar item where you can unmount and remount your external devices or server connections. I tried it with my kobo and it works, but only if I eject via Mountain – ejecting in Finder or Calibre makes it disappear as before, even Mountain can not see it then. I don't use Mountain myself, tested just with the trial version.)