First, yup there are still a few of us who play via Kong (via unity webplayer on FF-ESR or Pale Moon) and not sure that side is even being maintained, but....
For the last year or so there has been an intermittent problem where when trying to get in - all we get is the static image that says "Please create or sign into a Kongregate account in order to play". Now usually this resolves itself after a few hours but it seems to happen quite often to many people (according to chat) and happens to me a lot as well.
It turns out this issue is related to the page being served by cloudcade (kongregate.html) and it's use of the JQuery api. On this page the JQuery api is being loaded from ssl-webplayer.unity3d.com (https://ssl-webplayer.unity3d.com/downl ... ery.min.js) and this is being served by some round-robin edge content servers. Not all of these seem to have the JQuery API on them and some of them are returning a 404 error. This in turn is causing the game page not to load either the Webplayer or WebGL version of the game.
I managed to get it resolved personally by adding to my hosts.txt file the ip address of a different unity content server (rather than the one my DNS was returning) and the game came up with no issues.
Now I know the issue itself is actually Unity's problem, but you can link to a different copy of JQuery to fix this particular bug that has been plaguing us few Kong users for awhile.