Why didn't the One control Bombadil?
Tom Bombadil is an enigma in Middle-earth; we do not know what he is and his origin is never revealed. Tolkien did state in one of his letters that Tom had been created long before he thought about writing the Lord of the Rings and he put him in because he wanted an adventure for the hobbits along the way to Rivendell. For all that, Tolkien did use Tom to make a statement about the nature of power and domination. Tom is supposed to represent an existence whose key desire is in understanding ONLY. He is not concerned in using this understanding for any purpose. As such power and domination are completely useless to him and have no meaning or effect upon him: "He is master in a peculiar way: he has no fear, and no desire of possession or domination at all. He merely knows and understands about such things as concern him in his natural little realm. [He represents] the spirit that desires knowledge of other things, their history and nature...and entirely unconcerned with `doing' anything with the knowledge. . ."  And from the Council of Elrond: "Could we not still send messages to him and obtain his help? asked Erestor. It seems that he has a power even over the Ring. No, I should not put it so, said Gandalf. Say rather that the Ring has no power over him. He is his own master. But he cannot alter the Ring itself, nor break its power over others. . ."  The Ring cannot effect Tom Bombadil because he is outside the whole issue of Power and Domination; Tolkien uses Tom as an allegory that even this intense struggle between "good and evil" is only part of the whole picture of existence.
References  The Fellowship of the Ring  Unfinished Tales  The Fellowship of the Ring  The Silmarillion  The Letters of JRR Tolkien #131  The Fellowship of the Ring  The Silmarillion  Ibid  Ibid  The Letters of JRR Tolkien #121  The Silmarillion  Unfinished Tales  The Silmarillion  The Fellowship of the Ring  The Return of the King  The Silmarillion  The Fellowship of the Ring  Ibid  Ibid  Unfinished Tales  The Letters of JRR Tolkien #246  The Silmarillion  Ibid  The Fellowship of the Ring  Ibid  The Silmarillion  The Letters of JRR Tolkien #211  Ibid #153  The Fellowship of the Ring top