I've used an ancient Sony-Ericsson phone as WiFi to USB networking adaptor when WiFi didn't work on a laptop I was installing. That won't help you.
However a WiFi repeater (which can at worst case slow WiFi to less than 1/2 speed) can be setup with at least the existing details for the Kindle.
Some phones / tablets / laptops can either built-in via OS or an app be setup to be wifi repeaters.
Or you can use a laptop on ethernet cable at new site and have it be an airpoint with existing settings. May need an application if not built in to OS.
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