How do I explain the Trinity to my nine-year-old daughter?

Good luck. Part of the problem with the Trinity, and it is not a problem per se, is that explaining it becomes a problem because you never want to lean to one side or the other, and as soon as you do you are in heresy. It took the Church four centuries to figure this out. If you lean too far toward the unity of God you fall off the track, and if you lean too far toward the distinction of the three persons you fall off the track too, so the truth is in the middle, and then the hard part is explaining how you can have distinctions in a divine entity that has no distinctions because it is all the same substance. The best we have been able to do is by using names: three persons in one God. And there is a lot behind the word 'persons,' because if you are a person you have a personality, so the Son has his own personality, not the same as the Father or the Holy Spirit, though they share the same thoughts, motives, and goals. This is really the ancient problem of the one and the many, of universals and particulars, that separated Plato, who concentrated on universals, from Aristotle, who concentrated on distinctions, and no one has ever joined that gap. So all we can do as limited creatures is put labels on it; the Church does not know what the Trinity is any more than anyone else, and told us only what you cannot say: you cannot say three Gods, and you cannot say there are no distinctions. We will not really know it until we see God in heaven. So if your nine-year-old can absorb all that, more power to her; there are eighty-year-olds who have not reached that point.