1. The very first and the easiest tip to solve this problem is by restarting the network devices like modem and router. If all the other devices connected to the router are working properly, then your Fire tablet can be faulty. If your Wi-Fi connection is functioning properly, yet unable to access the internet or Kindle, then you need to go to 'Manage my Kindle account' to check whether there is any change in the settings.
2. After restarting the network devices, you need to restart your Amazon Kindle E-reader. Now restart the Router first followed by Kindle.
3. If required, then forget the network connection from the Kindle and re-add the Wi-Fi connection to the Kindle. Here you need to enter the password and SSID of your network. If this is auto enabled, then you don't need to add the SSID manually.