I believe Thunderbolt uses the exact same connector as mini-DisplayPort, so, as long as it is powered separately (i.e. isn't powered by the thunderbolt connector), you should be able to get use mini-DisplayPort to HDMI cable.
That's not going to happen on that build. The TB Display relies on an active PCI-e connection to function. If you had a cinema display this would be possible.
If you want this to work you're going to have to rebuild a good bit of your PC.