I've found a few articles to track the development of the term. Briefly, the phrases "penalty killing" and "penalty killer" originated in the 1940s and were widespread by the 50s. I didn't find the phrase "penalty kill" until 1984.
Here's an article from 1934 that shows the origin of the "kill" part of the phrase. In the 1930s, the phrase "killing time" was used, and could refer to playing defensively while shorthanded or with the lead.
March 16, 1934, The Granby Leader-Mail
Gerry Bates was off at the time serving a penalty and Aubin, killing time while his team was playing shorthanded, was about to circle his goal with Porter after him.
In 1939, coach Dick Irvin dropped the "time" part of the phrase and referred directly to "killing off penalties."
Saskatoon Star-Phoenix, Apr 8, 1939
"I'd like to have two men like Cooney Weiland and Gordon Pettinger of the Bruins just to kill off penalties," Irvin said.
In 1943, penalty-killing was used as a verb.
Ottawa Citizen, March 19, 1943
The Colvilles and Shibicky would have made it extremely hot for Claude Bourque in the home stretch had they not been so weary from penalty-killing.
And in 1944, penalty-killer was used as a noun.
Ottawa Citizen, Oct 24, 1944
Seldom lost to the team through injuries, (Syd Howe) has been extremely useful as a penalty-killer.
"penalty killing" and "penalty killer" have been a regular part of hockey vocabulary since 1945-1950.
However, I didn't find the phrase "penalty kill" used as a noun to refer to the shorthanded team's situation until the 1980s. In 1980, the variant "penalty kill times" was used to refer to the shorthanded situation.
July 16, 1980, AP
(Coach Eddie Johnston) said he wants the Penguins defense to be more offense-minded. He plans to fine tune scoring on power plays and improve the team's defensive stall during penalty kill times.
And finally in 1984, I found the first usage of "penalty kill" in its current form. It's in an article about the tactics and coaching of the penalty kill.
North Island Gazette, Feb 1, 1984
In penalty kill we will concern ourselves with the defensive zone and work from a box configuration. Space permits us to cover only one part of penalty kill, and for this reason we will stress the defensive zone.
The first system of penalty kill is referred to as the 'tight box'.
Later in the article:
The penalty kill remains structured around the four-man box system, however, the corner of the box closest to the box would be allowed to attack freely, pressuring the opposition into making mistakes.
The article sometimes used the definite article "the penalty kill" and sometimes did not; "penalty kill".
The Hockey Drill Book by Guy Palmer also referenced "the penalty kill" in 1984.
In the first figure, the team is shown split into two squads--one squad having only four players (on the penalty kill) and the other having five players (on the power play).
It's telling that both references were part of a discussion of coaching and tactics. Likely the phrase "the penalty kill", used to refer to the situation of being down a man and the players who play in that situation, arose out of coaching jargon and shorthand.