[Update]
I've been using a kongregate_shell.html file as is suggested on kongregate. With the Export for Kongregate option within Construct, this is unnecessary. If you export for Kongregate, use the index.html file for the iframe link in Kongregate.
Here's the reason why:
When you run the game through the kongregate_shell.html file, the api connects to kongregate_shell.html. Then, when your game attempts to connect to the Kongregate API, an error occurs because a connection has already been established. The first request then gets precedence and your game can never make use of the API meaning statistics, shoutbox, etc. will not get sent to Kongregate.
Here's another problem:
When run directly from index.html, kongregate_api.js still only connects occasionally. I think it has something to do with calling the API through iframes but I'm not sure. I'm still working out this issue. If anyone knows how to fix this, please post a reply.