George W. Bush --> Donald Rumsfeld --> Lt. Gen. Ricardo Sanchez --> Maj. Gen. Barbara Fast --> Maj. General Geoffrey Miller --> Col. Thomas Pappas --> Lt. Col. Stephen Jordan.
As you see, this bypasses Karpinski entirely. The ultimate truth, however, is that there were multiple chains of command, with many cooks in the kitchen from rival agencies, with Donald Rumsfeld leveraging his power to gather his own intelligence seperately from the CIA. He did this through a command chain that looks somewhat different.
Donald Rumsfeld --> Douglas Feith --> Michael Cambone --> Lt. Gen. William Boykin --> Maj. Gen. Geoffrey Miller --> Col. Thomas Pappas --> Lt. Col. Stephen Jordan.
There was also other forces in the government at work, applying often contradictory rules on how to deal with detainees. These chains included:
George W. Bush --> Condoleeza Rice --> Fran Townsend --> Lt. Col. Stephen Jordan
George W. Bush --> George Tenet --> CIA interrogators -->Pappas/Jordan
The CIA used Abu Ghraib as a place to interrogate prisoners "off the books", intentionally avoiding scrutiny of the Red Cross and other international organizations. Their methods of interrogation were not the same as those generally used by military intelligence, though it is unclear to what extent MI took its cues from the CIA operatives onsite and visa versa.
Basically, there were too many cooks, no clear chain of command... and Rumsfeld was at the center of all of it, playing several contradictory roles:
Official face of Pentagon: The US should abide by the rules of the Geneva convention in Iraq.
What he told his generals: We should toughen up the individual procedures to make our prisoners as uncomfortable as possible, in an attempt to get information from them.
What he told his intelligence wing: We should import questionable interrogation techniques used elsewhere, even if those techniques were used in a context where the Geneva Conventions don't apply.
Rumsfeld can't have it all three ways and still walk away with clean hands.
...and your hands won't be clean either, if you celebrate Yahoo's birthday by getting a free scoop of ice cream at your local Baskin-Robbins.
So, there you have it. Torture and ice cream. (Rumsfeld raspberry ripple?!)