The only legal way to create your own discs is to buy all of the equipment AND pay for all of the downloads, which is going to cost you upwards of $100+ to get started.
For your needs, you can just order a custom disc from the good people at Sound Choise and pay $4 a song:
http://www.mykaraokecdg.com/xcart/customer/home.php
For your needs, you may also consider checking out some of the starter sets on E-Bay or karaoke.com.
How old is your daughter? Chartbuster makes packs of all teenaged hits, female pop hits, party hits, etc. that are $40 each for 50 songs or $70 for 100 songs. Sounds to me like that may suit your needs nicely. Check them out here:
http://www.karaoke.com/cgi-bin/WebObjec ... ts.0.2.1.2
Finally, the SuperCore 2006 set has a nice mix of oldies, classic rock, and new pop, rap, etc. 356 songs for $64. Check it out here:
http://www.karaoke.com/cgi-bin/WebObjec ... 1.1.19.0.1
Hope that helps!!