As a last resort, try a full factory reset. This has helped for me and at least two other people, where the Kindle was stuck for (seemingly) no particular reason.
This will wipe all configuration and documents, so make sure you back up your documents first!
mntroot rw || exit 1
test -e /mnt/us/RUNME.done && exit 0
For reasons unknown to me, sometimes the Kindle performs a reboot after having run this, sometimes it doesn't. If it just hangs for, say more than 2 minutes, just hard-reset it (power button for ~20 seconds).
Hope this helps.