You could try to run
Kali Linux Live on a laptop off a USB drive (
see the official guides), set up a WiFi hotspot, connect your Kindle to it and then capture the network traffic on the hotspot interface with Wireshark.
You should be able to create a hotspot on Kali by going into network settings, create a new connection, choose type WiFi, set SSID to your desired network name, Mode to "Access Point", set IPv4 and IPv6 configurations to Shared, Security to "WPA2 Personal", set some password so random people can't connect to it while you're doing your thing, then save and activate that connection from the menu. These steps are correct at least for nmtui in a terminal, but I would expect NetworkManager GUIs to show very similar options.
Alternatively, I tried to write a terminal command that will create a connection like that, and it's as follows:
nmcli connection add type wifi con-name "Kindle Hotspot" autoconnect no ssid "Kindle Capture Hotspot" mode ap -- +ipv4.method shared +ipv6.method shared +802-11-wireless-security.key-mgmt wpa-psk +802-11-wireless-security.psk "YourPasswordHere"
Then you should be able to activate it with:
nmcli connection up "Kindle Hotspot"
You may need to add "sudo " before those commands if they fail due to a permission error.
You probably can't use WiFi to connect to the internet while doing this, so you'll need to connect with an Ethernet cable or tether via USB from any modern smartphone to get internet on the laptop.
Then when you open Wireshark you will get to choose the network interface to capture on, it's probably gonna be something like wl0 or wlp0s1 in your case, though you could capture on all interfaces and then filter by your Kindle's source IP address. The display filter (text field at the top of the screen after you start a capture) to show only DNS traffic is just "dns", you need to press enter after typing it in to apply.
You should start the Wireshark capture before you connect your Kindle to the network.
After you've done stuff on the Kindle that should trigger all of the relevant DNS queries, you have to stop the capture in Wireshark, then go to File / Export Specified Packets, use .pcapng format, in "Packet Range" select "All Packets" on the left and "Displayed" on the right (this will only export packets matching the "dns" display filter you set earlier). Select "Compress with gzip" in Compression options. Then you can save the file. You will need either another pendrive or to open Firefox and sign in to some cloud storage provider to upload the file though, as any files you save in the home directory on a default Kali Live system will disappear when you shut down the computer.
You can either go through the capture yourself and give me a list of hosts or share the packet export (not sure what privacy implications that has tho), either way works.
I haven't done this myself before, I did partially test the nmcli commands, and I've gone through the steps in Wireshark, but didn't test the whole workflow.
Good luck!