This is confusing because the letter that Kent is supposed to deliver is clearly intended for Regan, not Gloucester. And later [Act 2, Scene 4, lines 27-37] Kent says that he delivered the letter to Regan and her husband, the Duke of Cornwall, at their residence. After that, Kent goes on to say that Regan and her husband commanded him to follow them when they went to Gloucester's house. So Kent's journey ends at Gloucester's house, after all. For more about this matter see A.C. Bradley, Shakespearean Tragedy, NOTE U.