c. 1200, kene, from Old English cene "bold, brave, fearless," in later Old English "clever, prudent, wise, intelligent," common Germanic (cognate with Old Norse kænn "skillful, wise," Middle Dutch coene "bold," Dutch koen, Old High German kuon "pugnacious, strong," German kühn "bold, daring"), but according to OED there are no cognates outside Germanic and the original meaning is "somewhat obscure"; it seem to have been both "brave" and "skilled." Perhaps the connection notion was "to be able" and the word is connected to the source of can (v.1).
Sense of "eager (to do something), vehement, ardent" is from c. 1300. The physical meaning "sharp, sharp-pointed, sharp-edged" (c. 1200) is peculiar to English. Extended senses from c. 1300: Of sounds, "loud, shrill;" of cold, fire, wind, etc. "biting, bitter, cutting." Of eyesight c. 1720. A popular word of approval in teenager and student slang from c. 1900. Keener was 19c. U.S. Western slang for a person considered sharp or shrewd in bargaining.
mid-15c., perhaps a diminutive of Middle English helm (see helm (n.2)). But some sources suggest Old French heaumet (Modern French heaume), a French diminutive of helme "helmet," from the same Germanic source as helm (n.2); Barnhart writes: "Old English helm never became an active term in the standard vocabulary of English."