"Dangerous Astronomy" has a sing-songy feel because of all that repetition and rhyme, but it's also very casual and conversational, as if someone is recounting a story to a friend. This is particularly awesome, because the poem sticks to a strict poetic form while avoiding sounding too technical or poem-y.
The great thing about poems like this is their accessibility. It's the best of both worlds, like having a really laidback teacher recount a personal story that's full of complicated truths expressed in simple language. Not surprising, since Alexie is also a National Book Award Winning novelist and screenplay writer, so he probably knows a thing or two about how to tell a story in a captivating way.
Although the most obvious sound relationships are the end rhymes (rhymes that occur at the end of the line), Alexie also uses assonance throughout the poem. The hard "a" sound in "praise", "David", "baby", and "illuminate" sonically link words that reinforce thematic content of the poem. In other words, Alexie uses how the words sound, as well as what they mean, in order to communicate. Not easy to do as a writer, but amazing to experience as a reader.