The Kindle Fire is sold and supported by Amazon. ADE does not support
Kindles because Amazon chose to use a different version of software that is
not compatible with ADE. The library site is not correct if it is telling
you that you can download the library ebooks to your computer via their
Overdrive application, and then use ADE to transfer them to the Kindle.
What you can do is to contact the library and work with their technical
support people to figure out how to transfer the library ebooks formatted
for Kindles to your Kindle from the library site.
OK so finally worked out where I was going wrong. The Kindle wont link with ADE on a computer it has to be done on the device. Its right to say that regular Kindles do not support e-pub format but because Kindle Fires are web enabled this can be bypassed.
For anyone trying to use an Overdrive powered Library site with a Kindle Fire or Fire HD, here is what you do.
On your computer -
Log into your chosen Library site and set up your account. There will probably be a instructions and a link to down load the OverDrive reading app and Adobe Digital Editions. Download these and follow the onscreen instructions to register ADE with and Adobe ID. If you have a NON AMAZON e-reader, connect with a USB and it should show on the ADE screen as a device. (Remember ADE will NOT recognise Kindles)
If your were to check out a library e-pub book on your computer, it would appear in your ADE screen to be either read on screen or drag and drop into your connected device.
On your Kindle -
Switch on permissions to allow downloads from outside sites, switch on wi-fi
Use your web browser to download the OverDrive reading app from somewhere (Amazon and the Overdrive site wouldn't work for me, I had to get it from goodereaders.com)
When the app has installed, open it and tap the small square widget at the top of the screen. This will bring up a prompt page for you to register your Adobe ID.
Still on your Kindle browser, log into your library website and check out a book (there is a "get book" function in the OverDrive app but this would not find my library, possibly because I'm UK based.)
The book should now download straight to your device. Switch off w-fi when its complete.
Ignore your kindle bookshelf, go to apps and open OverDrive, the e-book will be there in a readable format.
For what it's worth, I loaded Aldiko on my original Kindle Fire & have been able to transfer books to it via ADE. However, based on *that* experience I bought my son a Kindle Fire recently, and after loading Aldiko and authorizing the device, ADE still doesn't recognize the new Kindle Fire. I didn't realize the Fires that they're selling now are the 2nd gen model, and they connect to the computer using a different transfer protocol mode - I think that's the issue. I'm guessing ADE doesn't recognize devices connected in MTP mode? That's the only difference I can see between my two Kindle Fires.
On OverDrive, the vast majority of books (at least for my library) is available in Kindle format, so that's not been an issue - I just check out the Kindle version. Recently a couple other local libraries added the Blio ebook service, though, and books are only in ePub or the Blio proprietary format. I was hoping to use the more popular ePub format, but applying your logic above, I think I probably need to unload Aldiko & load the Blio ereader on the 2nd gen Kindle Fire and use that format instead. That should work.
I don't know if you have found a solution for your question but I found this and it works for my kindle fire. I borrow books online and I have to have this to transfer the books to my kindle. It gives you step by step instructions. http://www.bluefirereader.com/blog/?p=158
Even if the Fire does not show up in ADE, you should be able to transfer files by 'sideloading'; that is copying the files from PC to Fire using a separate file manager such as Windows Explorer.
You may still need ADE to do the DRM bit of accessing the .acsm file and downloading the DRM .epub file. Then copy the .epub file. You should then be able to read it on Akdiko or Bluefire apps (both free) as long as they are properly registered with the same AdobeID. I never had any luck with using the Overdrive app to read bought books transferred this way.