The developers at Google had your sanity in mind when they created the YouTube Kids app. The app not only allows you to set a timer to limit the amount of time your child spends on YouTube, but you can also disable the background music and sound effects. Here's how:
1. Tap the lock icon on the bottom right hand side of your display.
2. Enter the number displayed on the "Grown-Ups Only" screen.
3. Tap Settings.
4. Tap the Background Music and Sound Effects slider so they're OFF / White.
This will only turn off the music and sound effects when a child is on the home screen of the app. Audio and music on all videos will still play.