I didn't think the DX used WiFi. My recollection is that it was a cellular network. AND ... it's possible that Amazon has stopped supporting that connection -- as I recall it was through Sprint which has been absorbed into T-Mobile.
My brother had a DX for a long while .... and at some point it just stopped connecting reliably. He gave it back to me to mess with and it would only connect sporadically for me. This was years ago. It's what decided him to just get one of the newer, though smaller, ones.
It's possible that it got itself de-registered when the battery was flat for so long and now the connection problem won't let you re-connect to the registration. Try toggling wireless on and off and restarting, but, other than that, you might be out of luck. The good thing is, if it's is showing as registered on Amazon, you should be able to download and transfer without any issues.
Maybe Amazon can help you out ... I'd contact them again via the web site. Here's a page that should work:
https://www.amazon.com/gp/help/customer/display.html/ref=hp_200127470_t1?nodeId=200375440You'll see at the bottom of the menu choices on the left is a 'contact us' button. You can initiate a chat or a phone call -- they'll call you.