I guess it's the contact clip: http://guide-images.ifixit.net/igi/UZ2ldiUKUbxvDJIv.m... This guide has info about this clip and where it should be placed (if not clear after the image above): iPhone 4 Contact Clip Replacement
What you should do is buy these: iPod touch (4th Gen) Adhesive Strips so you can glue your screen assembly back on the rest of the iPod. Make sure all connections are OK and everything fits so you don't have to re-open it again.
For children I would advice you to buy an iPad, this because of the larger amount of fun games for children and an interface which I find better for children. I think an iPad 2 will do the job and will be less expensive than one of the newer models. If money isn't a big problem then a newer iPad will be better.