Hm, 2 seconds is pretty intense! I have a HD60 myself and the delay is definitely there, but more around a half second to a second. Only in my situation, it is reversed with the game play happening on screen after I react to it, making a bit of an interesting video. But I find that it is acceptable in almost every situation. Unfortunately I can't think of a way to adjust for something that drastic if it is not in the settings. That's one of the drawbacks of that HD60, I find myself wishing I got the HD60S more often. Obviously I would say to just max out the delay as much as possible. Most audiences are pretty understanding that streaming is a difficult thing to get perfect.
If you're considering buying another part to solve the issue, Elgato makes a fancy thing called the camlink which is like a mini capture card for exactly what you're using your HD60 for. It's not quite as pricey as a full blown capture card so that might be up your alley.
Good luck!