After reviving a dead battery or after factory reboot Kindle DX can lose connection to amazon account. It's not possible to register it to an account, and synchronization doesn't work. It's possible to use kindle store and buy books, but they will never download.
I will answer with a fix caused by Kindle traveling back in time to year 1970.
To verify this just go to Menu and press Del to open search then type @time in the search bar and press Return. On the top you will see the date and time saved in the device. For me it was Jan 1 1970.

The problem is caused by incorrect date set on the device. Something in amazon server (probably https certificates) rejects requests from devices with incorrect date and no data is returned or processed.
Amazon support is not helpful with this problem.
In theory Kindle DX tries to update the date by calling
nc kindle-time.amazon.com 80(source) but that call currently returns an error so update fails. Kindle DX connects to internet only by using 3G so trying a "man in the middle" attack to fix the problem are impossible (without root access to a 3G cell tower)Fortunately it's possible to update the datetime manually. Unfortunately the latest software version of 2.5.8 that runs on KindleDX doesn't allow to change it from the user interface. Also command
;settime yyyy-mm-dd HH:MM(source) doesn't run on that version.The procedure as was in 2010 is described here. Currently it's outdated and following it in 2021 is misleading.
First step is to jailbreak your KindleDX. (You will lose warranty for 10 year old Kindle!) The software can be found here
The file needed is
kindle-jailbreak-0.13.N.zipI didn't have to use it because my kindle was already jailbroken by me some time ago so I'm skipping this step.Second file I used was
kindle-usbnetwork-0.57.N-dx.zipIt was enough to copyUpdate_usbnetwork_0.57.N_dxg_install.bin(dxg = Kindle DX Graphite) update the device "Home->Menu->Settings->Menu->Update Your Kindle" and then restart.This gave me to a new command that starts a Dropbear SSH server. I used the search field to run commands
;debugOnand then`usbNetwork(that character is backtick and not sigle quote!) now when we plug in the Kindle to a PC it no longer registers as a pendrive, but instead as a weird "Serial USB device (COM3)" device that windows 10 doesn't know what to do with.The only solution to this problem is to use a signed driver from file
kindle_rndis.inf_amd64-v1.0.0.1.zipfrom here After unzipping we need to right click ad "Run as admin" file5-runasadmin_register-CA-cer.cmdthis will accept certificates of the driver. After this it's possible to update the device driver from device manager by selecting the folder and allowing windows to find the .inf file.Now the device is known as
and it creates a new Ethernet adapter.
Then we configure this adapter to use IP 192.168.2.1 and mask 255.255.255
Now we can use putty to connect via telnet to the Kindle it's IP is 192.168.2.2 and port 23
And we have access to root of the linux installed on the Kindle.
Now it's enough to run
date MMDDHHMMYYYYlikedate 061512342021to set date to 06-15-2021 12:34 (source)After this procedure my Kindle DX Graphite was able to be re-registered (after factory restart) and it correctly downloaded some books from my amazon account.