Connect a computer to the router with an ethernet cable.
See the literature which came with your router for how to access the user setup screens in the router. Once in go to wireless security and review or renew the wireless security type and password. WPA PSK TKIP is the most common security type in use but check the manual for the Kindle. Make a note of the SSID or change it to one of your invention.
Go to wireless mode and ensure that it's set to a protocol matching your computer's wireless adapter and the Kindle's wireless mode (mixed mode is a safe bet). Check for a setting often called Access List that includes or excludes users by identifying the MAC ID unique to their wireless adapter. Disable that.
Disconnect ethernet cable.
Go to the computer's wireless networking setup screen and/or the Kindle and see if you can detect the router's SSID (wireless call sign). Select that network.
Apply the wireless security type and password to the computer's and the Kindle's wireless networking setup screens.