Hi Lilac,
Just checking in to see how your ant challenge goes. As I read the post I was thinking of what--besides anteaters--are natural predators for ants. Being overrun by ants like you are means something's out of balance in the system.
Here are a few natural predators of ants. Are any of these animals conspicuously MIA at your place?
Jumping spiders, lizards, snakes, birds--especially flickers, wrens, and sparrows.
Here's a link to a full list of ant eating critters. Hope this helps!
http://www.tightloop.com/ants/whatEatsAnts.htm