this is a fairly new computer, running windows 7 professional 64 bit, i've already checked my directx version which says 11, so i'm not sure why the dll isn't being found.
err your link is to directx runtime 9.0c, isn't that a older version then what i'm running?
I don't know what components of DirectX you already have installed on your computer, but you'll need to install the software I linked to,
since it's required to run Dolphin.