After setting up my two networked wifi-capable control boards, I just added the webserver functionality to the board that currently acts as the Access Point. This part was surprisingly easy. I just modified the code from this tutorial. When a button is pressed, a new client is found and the function that makes the brush draws a circle is called. When the button is pressed again (the state is off), the client disconnects. The webserver worked pretty much right out of the box.
After getting webserver to work, I simply added a flag that checks if the on screen button is pressed. If pressed, the webserver takes control. Else, the servo is controlled by the incoming UDP packets.