Try starting the game directly from the '..\SteamApps\common\Divinity - Original Sin\Shipping\EoCApp.exe' program file.

Did you try verifying local files for the game?

If that doesn't help, maybe reboot, shut down all non-essential programs, install DirectX again (sorry) from the '..\Divinity - Original Sin\_CommonRedist\DirectX\Jun2010' folder (perhaps install as administrator, unless the installer complains about that), reboot, and see if that convinces Steam that DirectX is already installed.