Damasio notes that skin-conductance responses are not noticed by the patient.  However, perhaps someone who is well-practiced in observing body sensations – for example, a very experienced meditator in the Burmese vipassana tradition  – would be able to discern such changes. I suspect so; in which case, a patient like Emily would be able to work around their recognition problem by noting when their skin sensations change. It’s known that they use workarounds; Emily, for example, “sits for hours observing people’s gaits and tries to guess who they are, often successfully.” 
Both Damasio’s theory of consciousness and Burmese vipassana place great importance on the interactions between body sensations and the mind. As I understand Damasio, he proposes that consciousness works like this:
1. The brain creates representations (he calls them maps or images) of things in the world (e.g. a face), and of the body itself (e.g. skin conductance, position of limbs, state of the viscera, activity of the muscles, hormone levels).Since I find pictures to be helpful, I’ve created a short slide animation on SlideShare that shows my understanding of this process; click on this thumbnail to go there:
2. In response to these representations, the brain changes the state of the body. For example, when it sees a certain face, it might change skin conductance; when it discerns a snake it might secrete adrenaline to prepare for flight, tighten muscle tone, etc. Damasio calls these responses “emotions”, which he defines as “a patterned collection of chemical and neural responses that is produced by the brain when it detects the presence of an emotionally competent stimulus — an object or situation, for example.” 
3. In a sufficiently capable brain (which is probably most of them) there is a higher order representation that correlates changes in the body with the object that triggered these changes. This second-order map is a feeling, in Damasio’s terminology: “Feelings are the mental representation of the physiological changes that characterize emotions.” ,  Feelings generate (or constitute – I’m not sure which…) what he calls the “core self” or “core consciousness”.
In Emily’s case, steps 1 and 2 function perfectly well, but the correlation between a face and changes in the body fails in step 3. The higher-order correlation still works for voices and body changes, however, since she can recognize people by their speech.
More acute awareness of body sensation might not just help clinical patients like Emily. In the famous Iowa Gambling Task, Damasio and Antoine Bechara showed that showed that test subjects were responding physiologically (again, changes in skin conductance) to risky situations long before they were consciously aware of them. A 2009 blog post by Jonah Lehrer includes a good summary of the Iowa Gambling Task and its results. Lehrer reports new research indicating that people who are more sensitive to “fleshy emotions” are better at learning from positive and negative experiences.
 This post is based on material in Antonio Damasio’s book The Feeling of What Happens: Body and Emotion in the Making of Consciousness (Harcourt 1999). Emily’s case is described on p. 162 ff. I also blogged about this topic in 2005 after reading “Feeling” for the first time.
 Damasio op. cit. , p. 300
 For example, vipassana as taught by S N Goenka. Other mindfulness meditation traditions also attend to body sensations (see e.g. Phillip Moffitt, “Awakening in the Body”, Shambhala Sun, September 2007) but the Burmese tradition places particular emphasis on it.
 Damasio op. cit. , p. 163
 Damasio, A. (2001) "Fundamental feelings", Nature 413 (6858), 781. doi:10.1038/35101669. Note that Damasio’s definition of “emotion” is narrower than usual usage, which refers to affective states of consciousness like joy, sorrow, fear, or hate. Damasio limits himself to the physiological changes which are more typically considered to be an accompaniment to, or component of, these mental agitations.
 Feelings so defined seem to correspond to what S. N. Goenka, a well-known teacher in the Burmese vipassana tradition, calls sensations, his preferred translation of the Pāli term vedanā, also often translated as “feelings”. There is some debate about whether vedanā refers just to sensations-in-the body, as Goenka contends, or to any and all pleasant, painful or neutral feelings such as joy, sorrow, etc.