I was able to download the 5.8.11 firmware for the Paperwhite 2 by using the direct link:
https://s3.amazonaws.com/firmwaredow..._v2_5.8.11.bin
That file isn't linked from the
Kindle Paperwhite (6th Generation) Software Updates at this time, but is probably coming soon.
I haven't tried links for other Kindle models.