Yes - you can download Libby ebooks to your Kindle Paperwhite!
Part 1 - You will need a 'smart device' to do the first part of the process. Use a desktop computer, Android phone, or Apple IOS device (ipad or iphone) to get the Libby app from the app store you already use on that device (Google Play, Apple App Store...)
Once you have downloaded the Libby App to your smart device, you will add a card - type in your Thousand Oaks Library card number and 4-digit PIN. In your Libby settings, please select the option to prefer Kindle ebooks. Some ebooks may not be available in Kindle, note that those titles will not be able to be loaded by your Kindle device.
With the Libby app, select and Borrow an ebook title, and then select the option to "Read now with Kindle." Your device will take you to the Amazon page to complete the download of your new Libby ebook title. (If your device is not already signed in to Amazon, you will need your Amazon login and password)
Once you are logged in to Amazon.com, you will see a yellow button to "Get Library book" tap the button, and the book will be checked out to your Kindle account.
Part 2 - open your Kindle Paperwhite device, and make sure it is connected to WiFi, the go to your "Home" screen. The new title should be visible on your home screen. If it is not, make sure you are on WiFi and synch your Kindle. The title should then show on the screen. Kindle device troubleshooting can be done by contacting Amazon Support.