The United States failed to join the Leage of Nations because Woodrow Wilson and his opponents refused to compromise on the wording of The Treaty of Versailles.
For Woodrow Wilson, the League of Nations was the best chance for long-term peace after the horrors of WWI. However, the Senate did not support the US joining the League of Nations. The said League of Nations was based on the Treaty of Versailles, which was not signed by the US since the government was not convinced by the conditions. The Senate refused to ratify the treaty that Woodrow Wilson had helped write and did not join the League of Nations.