Recover files from one computer on another computer

Ok I've got kind of a major problem here. Last night my computer froze up forcing me to improperly shut it down. After that the computer wouldn't start up anymore. It would freeze at the manufacturer's screen. After many attempts to access the bios I gave it up as a lost cause. I decided I needed to get my files off that computer which I've done numerous times before on Windows XP and using IDE HDDs. I usually set the corrupt HDD to slave on a secondary computer and copy the necessary files.

This time however I'm using Vista on both my machine and the recovery machine. The computer I can't access is a 64-bit. The computer I'm trying to set the corrupt HDD to slave is a 32-bit. I don't know if that matters or not. Both HDDs are SATA. Neither of them have jumpers. BIOs doesn't list the corrupt HDD, nor does it show up in disk management (duh). I can't change the BIOs settings either. All the HDD settings are grayed out.

Long story short, I just want to get these files off an SATA using a secondary computer that won't detect the HDD. All computers in question are running Vista. Any help would be greatly appreciated. Also let me know if there's any additional information that would be useful that I forgot to mention.