The readings "kin" and "kon" are on-yomi pronunciations for . The "kon" reading is the older one (go-on ) and "kin" is newer (kan-on ). They ultimately stem from Middle Chinese /ki̯əm/; notice that has the same on-yomi pronunciations.
As a general pattern, go-on pronunciations are somewhat less common (relative to kan-on) in everyday words and more usually found in words related to Buddhism. (But that is not to say Buddhist words only use go-on either.) In the case of , the kan-on pronunciation is more common (among the two on-yomi pronunciations) by far. You will simply have to memorise the exceptions.