That badge is for her not you.
I'm trying to keep my kid interested in music. He's only 2.5, but he seems to really like it. We have a no kids music policy, but I let him watch Jerry Garcia on YouTube all he wants! Our song of the week this week, however, has been Buffalo Soldier. I bought him a Ukulele, so hopefully eventually he'll want to learn it.
We used to do a thing where I would blow dry her hair and play a song super loud after a shower. We called it musical education. I tried to do the whole gamut of modern recorded music. Stuff I remember playing off hand is Glen Gould playing Bach, the Specials, Tracy Chapman Fastcar, led Zeppelin. She follows her own star music wise for the most part now a days. It's mostly garbage but she's 9. Her and her friends listen to the Velvet Underground a lot, which is cool but might not be good.
I dig the “song of the week” idea. But my kids were exposed to both children’s music/shows and everything we dig. The kids Watched the sound of music this weekend. Mary Poppins etc. they also loved Yo Gabba Gabba when they were 2.5 years old. Check that out. All the music is either composed by or selected by Mark Mothersbaugh. Lots of musical guests that you’d love. It’s a trip.