There is a box by Hauppauge that has an HDMI input and output that will allow you to record an unencrypted HDMI signal. It has a hardware H.264 encoder, which is particularly good for web streaming and use with most mobile devices H.264 does a great job of compressing the video stream. Unfortunately, none of the features specify live streaming capabilities, so I don't know if it will actually work for what you want.
The absolute best option would be to upgrade the gaming PC to an i7. It costs about $100 more than an i5 would (i5's are generally considered the best processor -- most cost-effective, that is -- for gaming purposes), which is less than the cost of a capture card that can handle a hardware transcode to MPEG-4/AVC/H.264 (the box that Hauppauge makes has an MSRP of $164). The upgrade from i5 to i7 gives you a slightly better core and adds HyperThreading technology, which goes a long way towards giving you a smooth encode for live streaming. Most forum posts I have seen have reported that an i5 isn't powerful enough to live stream a game that is being played, while the i7 processors are reported to handle live streaming comfortably. I only have anecdotal evidence, though, and nothing I have seen has ever gone into detail about what capture software or codecs were used for streaming, nor what games or graphics settings were required to have a successful stream.
Regardless of the lack of information I have regarding the details, I would suggest an i7-3770(K) unless you have the money and use for an i7-3930K (this CPU is an absolute money pit, but if anything can handle all that multitasking, this chip can). A separate Core 2 CPU just won't be able to handle a live encode at a resolution that you would be happy with. If you wanted to just have the C2D system record the stream, it would be perfectly capable, and then you could stream the performance later after the system had time to catch up with encoding the stream. You may be able to pull off a live stream of the event at a low resolution, but I am not completely sure about how possible that would be, nor how poor the quality would have to be in order to keep up.