I have been unable to download public library books to a gen 2 kindle, but books from amazon download without a problem. A search shows the book is downloaded but it remains in "now downloading..." indefinitely. Amazon CS couldn't help. Any ideas?
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You may need to use the "Download and Transfer" method using USB to move the book from your computer to your K2. It doesn't have WiFi, and some some books are too large to download over 3G. It may be that library books can't be delivered at all over 3G, I don't remember.
There have been many reports of downloading issues with these older devices. Many areas are dropping 3G support, which make it is problematic.
Look here, use the directions for the Touch model. Amazon seems to have removed help pages for the K2: