This is very helpful, thank you for pointing it out. I don't think I would have found this on my own.
oops, I'm confusing, the 12 volt kit with a 9 volt battery.
Yep, there is an equation for that. Go down to the table on the wiki page below, in the mAh column there is a number for each battery type. Alkaline AAA have a capacity of 1200 mAh (mA hours). Three batteries in series, like in your drawing / pictures, will also have 1200 mAh.
http://en.wikipedia.org/wiki/List_of_battery_sizes
Figure out the mA used by your LEDs.
(Battery voltage / resister ) x number of LEDs x 1000 = mA current used by LEDs
Example
(4.5v / 300 ohms ) x 10 x 1000 = 150 mA
Then you divide the battery capacity
1200 mAh / 150 mA = 8 hours