We do not know the exact dates on which Martin Luther King, Jr. worked on this speech. We do not know, for example, on what day he began to write the speech and we do not know on which other days he continued to refine it. We do know that he gave the speech on August 28, 1963. We also know that he was working on it up until the day that he gave it.
King gave this speech as part of the “March on Washington.” He gave the speech in front of at least 200,000 people, as he stood on the steps of the Lincoln Memorial in Washington D.C. He had prepared written drafts of the speech that he was to give, but it is said that he left his script and actually delivered part of the speech extemporaneously. In fact, it is said that the most famous parts of the speech, the parts in which the phrase “I have a dream” is repeated, is something that he had not planned to say and had not put in his written draft of the speech.
So, we cannot say exactly when the speech was written, but we know that it was never in its final form until he gave the speech on August 18, 1963.