The OP's solution is below. Adding as a community wiki Answer and removing the solution edits from the Question.

Found the Kindle using the sniffing technique described below. However I'd still like to know if there's a way to find it that's less overkill.

I discovered that the Kindle updates its contents daily while in deep sleep around 5:30-6:30AM. So I set up the delivery of a biggish pdf file through the kindle email so I'd ensure a long enough connection, end set up three laptops with a Kali Linux running live with airodump sniffing packages (modified from here: http://www.androidauthority.com/capture-data-open-wi-fi-726356/ )

The following procedure can be used to physically find any active wifi device connected to a network:

Three lines of code in the terminal in at least three Kali Linux running devices (2D location, at least 4 receivers out of plane for 3D):

To start monitor mode on a wifi card (in this case wlan0, check on iwconfig if needed):

airmon-ng start wlan0

Airmon usually changes the card name to wlan[n]mon after this. Check the wifi interface name on iwconfig and modify accordingly if the next line returns a related error.

Then run airodump to find your wifi network and its channel:

airodump-ng wlan0mon

Enter Ctrl-C to end the capture once you note the desired network's channel

Run the following line to run airodump again only at the specified channel and record the data to the "allthedata.csv" file in the terminal directory (usually the root for a live cd/usb if you didn't change it). Replace the square brackets by the channel number:

airodump-ng -c [channel number 1, 6 or 11 usually] -w allthedata wlan0mon

I started the pcs before going to sleep and as I woke up a bit past the usual update time, I hit ctrl-C to stop the capture and looked up the device on the allthedata.csv files by its MAC number which can be obtained at the router logs. There are in fact 4 allthedata files with different extensions per capture, and on the xml you can find the manufacturer of the device in the logs, so even if you don't know the mac address by looking up an Amazon branded device connected to your router you may find the Kindle entry.

At the Kindle entry, find the signal strength value and note it for all the receiver computers. Then convert the values from dbm to mW using the following formula: mW=10^(dBm/10)*1000.

To triangulate the signal (for a 2D location), draw a chart with the position of the laptop wifi receivers and for every pair of receivers, draw a line perpendicular to the line going through the points in a point inversely proportional to the square root of the mW power values, as the signal drops of proportionally to the square of the distance (for 3D use at least 4 receivers and a surface between the points).

The lines should cross at the source of the emission - the Kindle was at a closet in this case. The margin of error was about 1m/3ft even with several concrete/brick walls in between the receivers and the emission.

Notes: