If you log into the Router, you can see the SSID name that your router transmits.
You can also go down to Attached Devices and see if your router is seeing your Kindle.
If the router doesn't see the kindle as an attached device then you need to go into Settings on your Kindle and make the Connection.
1. Tap Settings 2. Tap Wireless & Networks 3. Ensure Wifi on checked 4. Under wifi networks you should see a list of WiFi routers available in your neighborhood 5. Tap on the SSID of your router 6 . Enter the wifi password from your router and tap Connect