What factors allowed the British to enter the war confident that they could beat the American colonists?
The British, at that time, was one of the leading powers of the world. They had (or had once had) colonies in South America, Central America, the West Indies, Asia, and Africa. They outnumbered the American colonists, they were wealthier and more powerful (they could hire mercenaries), and most countries were afraid of them.
France did not side with America originally because England was too great of a power. France declared war on England in 1778, not earlier, even though France and England were great enemies.