Burger King once had a name that was really an advertisement for a kitchen appliance.
The chain began as Insta-Burger King, founded in Jacksonville, Florida, in 1953. Wikipedia gives July 23 as the date. The "Insta" came from the Insta-Broiler, an oven the founders had bought the rights to. Wikipedia says it cooked 12 patties at once in a wire basket, grilling both sides. None of the sources I read defines "Insta," so the page ties it only to the oven's name.
The founders visited the McDonald brothers' restaurant in San Bernardino, California, before opening. Sources spell the first founder's surname both Kramer and Cramer, so this page does not rely on either. The other founder was Matthew Burns.
In December 1954, two Miami men, James McLamore and David Edgerton, opened a franchise at 3090 NW 36th Street. They found that the Insta-Broilers broke down under heavy use. According to two accounts, Edgerton and a mechanic named Karl Sundman built a replacement broiler in about three weeks. Wikipedia says only that McLamore and Edgerton developed a flame broiler.
Once the new broiler replaced the oven, the "Insta" no longer described anything. The parent chain, which had grown to more than 40 locations by 1955, was struggling financially. In 1959, McLamore and Edgerton bought it and renamed it Burger King.
The sources disagree on the date of the change. Wikipedia gives both 1954 and 1959, and the more detailed passages point to the 1959 purchase.
No source I found says why the founders or the later owners picked "King." The simplest reading is a royal title meant to sound big. That is an inference, not a documented fact.
The other half of the name has its own history. Etymonline says hamburger meant a person from Hamburg in the 1610s, and hamburg steak is recorded by 1880. The shortened burger is an American form attested from 1939.