First of all, you should realize that the term "detente" is not typically used in connection to the Korean War. We do not use the term to refer to anything that is the opposite of "brinkmanship." Detente refers to a time, typically given as 1969 to 1975, in which tensions between the superpowers were generally low.
Second, if your essay is about ending the Cold War, Korea is not a great example to use. This was the early stages of the Cold War. The end of the shooting war in Korea did not do anything to end the Cold War itself.
So, it sounds more as if you're trying to argue that negotiations and less aggressive actions were more effective than brinkmanship in solving problems during the Cold War. If so, the Korean War can be of help. Brinkmanship can be seen in the actions of people like General MacArthur. His push to the Yalu did not end the war but rather brought China into the war. Threats of the use of nuclear weapons did not help either. The war only ended when military stalemate was achieved. Once this happened, the sides had an incentive to back down and "peace" could be achieved.
The Korean War did not end. The two sides stopped fighting, but it is still an active war.