Question: While walking I experience an aching, knotting pain throughout my almost non-existent arch, but nothing seems to help.
Answer: While I can't diagnose your problem—and I suggest that you consult a podiatrist if you have not already—I can make a couple of suggestions.
Number one: Are you buying your walking shoes long enough? Many, many people persist in buying walking shoes that are too small for athletic activity, which can cause foot problems. Your foot swells as you walk and you may actually cramp your foot trying to retract it inside the shoe to avoid rubbing and blistering. I convinced my own sister to switch from a size 6 to a size 8, and it relieved her foot and knee pain! You should be able to press your finger down fully in front of you longest toe.
Second, check yourself when you are walking briskly. Are you tensing up your lower shin and foot in an effort to increase speed? I often do this and it's counterproductive. As you speed up, imagine your feet are like feathers or roller blades—--any image that helps you to keep the foot and lower leg relaxed as you roll through and push off. See if that helps.