I ran into a similar problem when I got my first iPad and configured it by restoring my iPhone's image into it.
Assuming that iPhone functions like telephoning and cellular data connectivity are OK, it is likely that some/many of the applications that transferred from the iPod are not compatible with the iPhone. For each application that does not work well, you will need to delete it and add it back in again. Since you used the same Apple ID on the new iPhone as on the old iPod, any paid apps are still paid for and the appropriate iPhone version will be installed.
This may be laborious, depending upon how full your daughter's iPod is, but it is a one time pain for long time joy. You should also feel comforted by the fact that initializing the iPhone in this way took care of the lion's share of the setup effort. Good luck.
Edit: One more thing. Be sure to go to Settings | General | About on the iPhone and change the "Name" to something different from the iPod's name.