Jastrow defines כיח as "cough, phlegm," and lists its root as כחח.
I would suggest that כיח is of imitative origin, an onomatopoetic word, based on the sound of coughing up phlegm or clearing one's throat of phlegm.
Indeed, the English word "cough" appears to be of imitative origin. From Dictionary.com, the etymology, according to Random House Dictionary is:
[Origin: 1275–1325; ME coghen, appar. < OE *cohhian (cf. its deriv. cohhettan to cough); akin to D kuchen to cough, G keuchen to wheeze]and American Heritage Dictionary gives:
[Middle English coughen, ultimately of imitative origin.]The gh sound in Middle English is similar to the ch sound of khaf, כ. While Hebrew כחח or כיח has ח in place of כ, and they are not the same sound, the word might be of similar origin. That is, both may have been imitative.
Balashon, what say you?