I don't know LA all that well, but Griffith Observatory, CA Science Center, and La Brea Tar Pits definitely are relevant, and I think a 14 year old will love them (if she likes science at all). Allow more time for CA Science Center than the other two because there is a lot to do there. If you want to look through the telescope at Griffith, get in line early. They will stop letting people in, even if you've been in line for a long time, once they hit closing time. But there is plenty to see and do even without looking through the 'scope. Be sure to visit their website because they have a number of special events, many of which are free, such as viewing events on the plaza.
Do keep in mind that the weather is fickle. Be sure to have layers because it can get unexpectedly cool even in the summer.
DS '04 "Boogaboo"
DD '08 "Lilybear"