I just happened upon this four-years-old thread right now, while looking for something else. The person who was asking might have registered just to pose the query, and may now be long gone from these lands. But better late than never.
My guess: the singer in question was not Peggy Lee but Peggy King. It would not be the first time that the two Peggys get confused. I've seen as least one write-up that refers to Peggy Lee as "pretty, perky Peggy," which was a phrase applied back in the day to La King, not to La Lee.
Once, I bought a Stan Kenton CD that in which Peggy Lee was supposedly heard, singing various standards with him. Imagine my frustration when the singer turned out to be Peggy King. (Nothing against King, whose singing I like well enough, but I was unhappy to find my expectations dashed...)
Back to the query itself. I don't know if Peggy King ever sang in Jack Benny's radio program, but she certainly appeared in his TV show. You can find YouTube clips of an appearance in which she isngs "When Love Walked In, Brother."
Ivan