I don't know about much about how much a 9 year old girl could or should do to develop the type of serve that the pros are using.
This video, taken from behind, shows clearly the rapid
internal shoulder rotation that provides the greatest contribution to the racket head speed at impact. Look at the rapid rotation of the bones at the elbow. (ISR is upper arm (humerus) rotation).
https://vimeo.com/27528701
To do stop-action single-frame on Vimeo press the play-pause control as fast as possible.
Notice 3 angles at impact:
1) The angle between the straight arm and the racket. This angle is to get the very high racket head speed from the axial rotation of the arm. See Toly link.
2) The wrist at impact - it appears to be in a reasonable position not near any limit of the wrist joint's ranges of motion.
3) Notice the angle between line between the two shoulders and the upper arm. One shoulder up, one down, the line between the shoulders and upper arm are somewhat in a straight line, say, roughly, 170 d.. This orientation minimizes the risk of shoulder impingement. This issues has been explained in detail by Todd Ellenbecker in a Tennisresources video - a very important issue for long term shoulder health when playing tennis. It's dangerous for your daughter to work on her serve if you and she are not aware of the information discussed by Ellenbecker. Search: Ellenbecker shoulder video
I would recommend that you do high speed video
from behind - in about the direction of the ball's trajectory and look for these important angles as shown in the above video and others in that Vimeo collection.
For reference, the part played by internal shoulder rotation in the tennis serve has been described by biomechanical researcher, Bruce Elliott. On the TW site, Toly has some nice posts describing the serve and especially the angle between the arm and racket at impact.
http://tt.tennis-warehouse.com/showthread.php?t=361610
There is a thread on the physics & science of 'pronation'.
http://tt.tennis-warehouse.com/showthread.php?t=370729&
A great and very readable reference on the tennis strokes is Technique Development and Stroke Development (2009). Bruce Elliott. Available through the ITF. It has several discussions of what developing young players, given their strength limitations, are capable of doing at various ages.
https://store.itftennis.com/product.asp?pid=86&previousscript=/home.asp