The idea of the model 19, aka "combat magnum" was to make a smaller, lightweight 357. The K frame is the S&W medium or 38 frame. The I frame, which later became the J frame was a 32 frame, and the N frame was the 44 frame.
The first 357s were made on the N frame, because that was the strongest and heaviest frame S&W had until the X frame (460 and 500 mag). Approx 20 years after introducing the first 357 magnums, S&W saw a market for a smaller, handier 357. With a few alterations, they got the 357 magnum to work in the K aka medium frame. There were several K frame 357 magnums: the model 19 (adj sight, blue or nickel), the model 13 (fixed sight, blue or nickel), the model 66 (same as 19, but SS) and the model 65 (same as 13, but SS). They were very successful, for many reasons. They were one of the lighest 357s on the market. They were lighter than any design that I know of from Ruger (maybe until the SP101), Dan Wesson, or Colt (until the Magnum Carry). They were also smaller than many of these other models.
Later, issues arose with the model 19. In my opinion, two factors caused these issues, lack of basic maintenance AND light fast 357 magnum loads (125gr at 1400 fps). The guns were said to eventually shoot loose (who knows how many rounds that would take) and many model 19s ended up with cracked forcing cones. The 6pm portion aka bottom of the forcing cone was thinner, because the cylinder window space was was not tall enough to make the forcing cone competely round, and the same thickness throoughout. Many K frame 357s were fine despite the design, while others ended up cracking there. Usually the cracked ones were fed with 125 gr loads which were fast, AND they were not cleaned well. Usually if a particular gun had one or the other in its history, it was fine, but both combined was linked to the failure. This failure, and the complaints about the recoil and shooting loose, spawned the development of the L frame, which had a thicker forcing cone, a full lug (initially) for added weight, among some other improvements.