Can you play a reaction to an event, if that event is the thing that let you draw the reaction?
Would she be allowed to play the card in response to losing this conflict?


According to my analysis, yes.

Seems that The Art of War has an interrupt effect, which basically means that it resolves before the other effect or condition that triggered. From the rulings on interrupts (p.10):


  An interrupt is a triggered ability whose text is prefaced by a boldface “Interrupt:” precursor. An interrupt ability interrupts the resolution of its triggering condition, sometimes canceling or changing the resolution of that condition. Always resolve interrupts to a triggering condition before resolving the consequences of the triggering condition itself.


Therefore, your opponent drew those cards before the Military Conflict came to a resolution. This enabled her to be able to use the reaction from her card, as "it was there" before she "lost the military conflict".

This last sentence is important, as Reaction effects require that it "may be initiated only if its specified triggering condition occurs." (also from the rulings, p.13). If The Art of War were not an Interrupt, it will require for the Conflict to end in order to take effect, situation that will result in the card being pulled out to lose timing and not be able to activate (as the last thing that happened was the resolution of the last effect, and not the end of the Conflict).