The K2 uses cell towers for a connection, not WiFi. Depending on the version, it used Sprint or AT&T. My understanding is providers no longer supper the slower 3G connections.
It's also my understanding that books already purchased can be downloaded, but you can't access the store anymore. Of course if there is no 3G connection, nothing will work for wireless download directly to the Kindle.
The "download and transfer" option will still be available.
https://www.amazon.com/gp/help/custo...e%20(5th%20Gen).