The flash chat plugin creates an irc socket connection to chat, and relays that back over a JavaScript interface. Unfortunately that’s being prevented in your embedded browser.
You might want to look into using https://code.google.com/p/chromiumembedded/ with the Chrome’s Pepperflash plugin, as that’s the only way I’ve seen possible to get an embedded browser to work with Twitch chat. The reasoning is probably due to sandbox issues caused by it being an embedded framework, and Pepperflash is a native Chrome plugin (and as far as I know it has native hooks).
Unless Twitch switches to something other than flash, I am doubtful you’ll be able to get it to work without trouble.