I have sometimes lost connection -- though the wifi still seems to be on. Sometimes I've been told it couldn't connect; sometimes it just hasn't d/l'd books that I know it should have. In those cases (after confirming my home network does see the internet) I just toggle wifi off and back on, it remembers the password, reconnects and all is well.
Occasionally that doesn't work but a full restart does.
If a restart doesn't work, I assume it's an Amazon problem; I just wait a bit and try again.
But I've never been asked to re-enter my password unless I'd just reset it to factory state for some reason, or had told the kindle to forget my network first.
Of course, if MY router isn't getting out to the world, that's a different problem. I've not had Kindle prompt me for a password in that case, it just connects and nothing happens.
If the modem shows signal I do the old "unplug it all and then plug it back in" thing. That usually works.
If there's no signal at the modem, well
there's your problem! 
If temporarily unplugging the modem doesn't trigger anything I contact the ISP provider and usually hear a recorded message that there's an outage in my area.
I kind of feel like if your kindle keeps asking you for the router password then either you haven't entered it correctly or you have a problem with your Kindle or your router.
I also think it's possible that Kindle doesn't always play well with older routers. I know some android based devices have trouble staying connected to older ones, depending on the security it's using.
But, as I've never actually experienced that, maybe I'm mistaken. I admit I don't know nearly as much about this sort of thing as Morf. For me, if it works, I'm happy and I don't look further unless it STOPS working!

Then I try the steps above. If none of them work, I seek professional help.
