Not sure you are going to find an integrated with good sound. Nakamichi used to make systems like that, but they are pretty much gone.
How about a PC or Mac, optinally a Dragonfly usb DAC and Audioengine powered speakers? You can even drive the speakers from the headphone jack if you are OK with the DAC in the PC/Mac. You can use iTunes or some other software. For streaming, the family could use Pandora or something like that. Also gives you Internet radio. More money that Elizabeth's solution, but also more flexible. Not an All in One but pretty compact. You could also use the Audioengine wireless connection if you want to locate the source in a different place.
For a more conventional system, the NAD 715 is a nice little CD/Radio integrated. Use with speakers of your choice. Maybe add an iPod dock or use a usb memory stick. Of course, all the volume players make a $300 system, of varying quality.
Good luck.