Would Britain still control America had it not been for the failure of the three-pronged attack plan led by General Howe?
The most obvious answer to this question is no based on the history of the British Empire. The British maintained control of most of their colonial possessions until the end of World War II. By that point they had granted more indpendence to most of their colonies.
Some of their countries that maintain closer relationships for an extended period of time. Australian was a part of the greater empire until 1986 when they were granted soverign status. In 1982 Canada was not longer under the control of the British. Since these two nations are similar to the U.S., it is not too much of a stretch to assume that the U.S. would have enjoyed some sort of similar treatment and been granted independence in the 1980's