One of Koch's collaborators, Francis Crick (yes, that Francis Crick, much later in his career), put forth an interesting theory with Koch that while perhaps is a bit far fetched, it's worth mentioning for sake of a slightly different perspective.
Crick and Koch posited the claustrum (see diagram below) as one of the seats of consciousness in the brain. Koubeissi, Bartolomei, Beltagy and Picard (2014) supported their hypothesis when conducting electrical stimulation mapping on a 54-year-old epilepsy female patient, who became unresponsive under the stimulation in the left claustrum/anterior insula region. However, several limitations in this study need further investigation, such as a high electric current, 14mA and lack of stimulation on right hemisphere.
As you've already read some of Koch's work, you have some idea of their working definitions for consciousness, but in brief
...almost all neuronal theories of consciousness...need...continuous interactions
among groups of widely dispersed pyramidal neurons that express themselves in the ongoing stream of conscious percepts, images and thoughts.
Now, before you ask how a small strip of gray matter running along between two prominent white matter tracts could essentially "bind" together stimuli from the entire brain consider the following points (drawn mainly from non-human primate and cat data):
The type I cells of the claustrum receive inputs from nearby areas of the cortex, and also have been found to project back
There is significant overlap between the representation of sensory and motor cortices, visual cortices and prefrontal areas, along with motor cortices and prefrontal areas within the claustrum
Interneurons there may possess a more "finely-tuned" timing sensitivity, appropriate for binding together the areas responding to multimodal stimuli
The cat claustrum, in addition to the connections with sensorimotor areas and the visual cortices, also has a prominent section ventral to the visual projections that integrates auditory information
Crick and Koch assert that these points support the idea that the claustrum might be acting as a (orchestral) conductor for multimodal stimuli. Using gap junctions (direct connections between cell membranes, in this case being used as high-speed electrical synapses), interneurons of the claustrum could be employing the type I cells to "grab" and piece together information from disparate portions of the cortex simultaneously.
Of course, once the information is bound together, it would remain to be seen as to which particular structures would interpret this bound data. Crick and Koch don't really comment on this, but since there are strong bidirectional connections with the prefrontal areas, perhaps the "conductor" is also sending back cues upon which the attention of the frontal lobe could be gated, but that is simply an educated guess, and I have no further support for that.
So, as far-fetched as it may seem, a brain structure that is small on volume may have a significant enough representation of cortical information, ability to project back and "conduct" cortical areas, along with an interneuronal backbone capable of precise timing, all of which give it some chance at being an important seat of consciousness in the brain.
Crick, F.C., Koch, C. (2005) What is the function of the claustrum? Phil. Trans. R. Soc. B 360: 1271–1279 [DOI] [PDF]
Koubeissi, M. Z., Bartolomei, F., Beltagy, A., & Picard, F. (2014). Electrical stimulation of a small brain area reversibly disrupts consciousness. Epilepsy & Behavior, 37, 32-35. [DOI] [PDF]