For those of you wondering: I Googled "how to kill winged ants," and came across an article that suggested a spray bottle of soapy water if you didn't want to use aerosol ant poison. The same article also suggested that time was of the essence when dealing with winged ants (like, to the point that you shouldn't even leave the room after discovering them). So I grabbed the nearest spray bottle, which happened to contain that Clorox Green glass-and-surface cleaner stuff. And we spent the evening spraying the swarms. After doing the initial spray, I tasked Jeremy with mopping up the massacre (times like this, it's particularly useful to be married). He had to spray a few times; the last about 15 minutes before bed. I
tossed and turned, thinking about the ants behind the drapes (J, of course, slept like a baby - how do you guys do that?). This morning I pulled back the drapes, got down on hands and knees and peered under the radiator. Nothing save a handful of ant carcasses. I figured J had mopped them up one last time before bed, but apparently he had not. Those ants had come in and carried off their dead like you always hear they will do!
Hopefully they will pass the word on to their buddies that our bedroom is a hostile environment. We can hope, anyway.