Q: Can not install any apps on daughter ipod

Why can't you download apps? What is happening when you try?

First try holding the sleep/wake and home button until the Apple logo appears and try again.

If you have ask to buy on did you approve the purchase? Request and make purchases with Ask to Buy - Apple Support

Were you asked for daughter's iTunes Apple ID and password?  You should have been asked.  If not, log daughter out of iCloud, wait 10 seconds, sign back into iCloud and try app purchase again.

This is how it should work and works for me.

Make a request with Ask to Buy

If Ask to Buy is enabled for your account, browsing the store is the same as always, but before a purchase is complete, a request will be sent to your family organizer. Just use these steps to send a request to your organizer:

Tap the price on the item that you want to get.

If asked, enter your Apple ID and password.

Tap Ask. Your request will go to the organizer. You can tap "approve it in person" if the organizer is nearby and they can make the purchase directly from your device. Even if you make multiple requests for the same item, your organizer will get only one notification.

Do you have it set up to ask to buy?

To sign out of iCloud,  Settings > iCloud and scroll to bottom to sign out.  Ignore the sign out warning.  It won't harm the iPod.  Then, when signing back in, merge data.

I know you've restarted the iPod recently, but I would do it again and make trying to sign out of iCloud again the first action after restart.