"Staticness" of IP address depends on either your router configuration (if you use SSH over WiFi) or it would be always static if you use SSH over USB.
BTW, WinSCP works for me any time I use it. It's just Kindle not re-indexing its storage without some nudge (the one I'm aware of is plug it as USB drive and then unplug) if you want to use it to put books there...
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