If the WiFi is goosed not worth repairing, can get them in Cash Converters etc for £30 long as you don't mind it being an old model.
Only other thing that I thought of and it's a long shot is that your router is only broadcasting 5GHZ 802.11n or ac which the Kindle can't see as it's only 2.4GHZ 802.11g or b, but that's highly unlikely as it would still pick up your neighbours.