Here are two definitions*:
"down for the count:
Defeated, finished, doomed, as in I doubt if he'll ever leave the hospital; his illness really has him down for the count This expression alludes to the losing boxer, unable to get to his feet before the umpire has counted to ten (the count). [1920s]"
"out cold :
Also, "out for the count," "out like a light." Unconscious; also, asleep. For example, He crashed into the wall and was out cold, or Willie punched him too hard, and he was out for the count or Don't call Jane; she's out like a light by ten every night. The adjective cold refers to the lack of heat in a dead body and has been used to mean "unconscious" since the second half of the 1800s. The first variant comes from boxing, where a fighter who is knocked down must get up before the referee counts to ten or be declared defeated; it dates from about 1930. The last variant alludes to turning out a light and dates from the first half of the 1900s."
In your example, only "out like a light" seems appropriate.
*The American Heritage Dictionary of Idioms by Christine Ammer. Houghton Mifflin Company. 1997