On your Kindle Fire, while connected to the Internet, navigate to the Settings -> More -> Device menu and turn on "Allow Installation of Applications from Unknown Sources." Click OK when a warning appears.
When the download is complete, click "Install" when asked if you want to install the app. If the install page does not appear, click your Kindle's name in the upper right corner, find the app, and tap it to bring up the Install page.
Now that you have a compatible reading app installed, follow the basic iPad instructions to find library ebooks and download them into Bluefire Reader or a similar app.
If you wish, you may now turn back off the "Allow Installations of Applications from Unknown Sources" setting which was changed in Step 1.
Reading Library E-Books on Other Kindle Models (NOT Kindle Fire)
Unfortunately, older Amazon Kindle devices do not support the downloading of ebooks from academic library platforms.
If your Kindle device has a web browser, then you may access most of the library ebooks on the web for online reading while connected to the Internet.
You can also download PDF documents from certain library databases or other sources and send those PDFs to your Kindle. Read the complete steps to send a PDF to your Kindle (instructions provided by an external site).