We had similar issues with Smart TVs and the fix for that was to change the DNS.
Did you try setting the router to Google DNS?
Here's how -> https://developers.google.com/speed/public-dns/docs/using
Given the details in your post I can't offer much. There are other reasons but here it works fine.
I have had a kindle keyboard with wi-fi for a year and a half. Two months ago, it stopped downloading books from Amazon. For two months, I had trouble-shot every possible fix available. I've worked extensively with Amazon and Verizon to get this problem fixed. According to the forums I've checked, this is a common problem. Sometimes the problem resolves itself, but for me it hasn't. Does anyone have any ideas as to what is going on, and ideas on how to fix my kindle? I just got a replacement, and it, too, does not work. Like my original kindle, it can access the internet, and Amazon, but will not download the books. Both Amazon and Verizon claim deniability, since everything else seems to be working. If I get a Nook, will this also happen?