Need a low-power wifi 'switch'

I got a off-the-shelf circuit (china) that runs off 12v, and will turn on / off a switch. Works great, AND... it drains my battery! It draws 30mA continuously. I need to be down at maybe 1 mA average power draw.

I don't mind laying out a little circuit with the relay/regulator/etc. But I'd like to get a ethernet 'module' that I can just solder to my circuit. Trying to find one is making the back of my eyes hurt!

So, any recommendations? Nothing fancy needed, it's not a life-critical switch, just something to make life a little easier. Basic wifi, not really worried about one band vs another. Thank you in advance!