I agree with all of you who say that the only possible emotion KC could have shown that would have been incriminating is anger. If she had simply showed anger at the news, that in itself would be incriminating because there is really no other emotion that could be construed as totally self-centered, unfeeling and remorseless.
However, if, as some of you suspect, she was directly her anger specifically at JB, that may also be a reason he would fight so hard to get this tape away from "Enquiring Minds" - it would show he probably had knowledge of her guilt and that she was blaming him for something he had assured her somehow would not ever happen.
We also have a third possibility - although it probably would not have played out if KC was observed behaving angrily - and that is, as has been rumored in other places, that JB once again may have been physically behaving in a way that would be considered highly inappropriate with his client, perhaps offering consolation in a way that would suggest a previous level of physical intimacy that went way beyond the boundaries of simple client-attorney relationship. He's had his hand slapped more than once for that one, and in this situation, with all its implications to them both, he may have just taken it over the edge.
I can see him fighting hard to suppress any or all of the above scenarios.