Matthew, the reason a kite stays airborn is usually because it is not streamlined so the flow becomes turbulent in the wake (ie static pressure is not recovered on lee side). If air was inviscid then a kite would fall to the ground and also there would be no parasitic drag! Unfortunately the lack of coanda effect would mean our precious helis would not get aiborn either...
Assuming kite flies as an aerodynamic wing: The flow over the kite requires wind so the total stagnation pressure is higher than atmospheric (static + dynamic pressure). The kite induces vortices which by various methods (either magnus effect or vortex shedding) speeds up flow above the kite. This increased local velocity means an increase in dynamic pressure hence a reduction in static pressure to keep total stagnation pressure constant. This is why a wing produces more lift at higher airspeed.
Don't forget there is always a net change in momentum. I have a nice video somewhere of some Airbus wind tunnel work. Once the model flies through the smoke field the wingtip vortices are generated as expected. Afterwards the vortices slowly work their way to the ground, as each vortex causes a downwash on the other.