It turns out ladybugs are like Roombas. They scoot around in a haphazard fashion around the plant, only munching on aphids that their tiny little heads run into. A little off to the side and the ladybug simply steps on and over aphid. Also like Roombas, aphids get full easily. In desperation, I resorted to picking up stray and hungry-looking ladybugs and placing them on aphids to get more aphid-eating to happen.
After a week had gone by, there were no more ladybugs and tons of aphids, mostly on the undersides of leaves. Frustrated, I decided to escalate to chemical warfare. I got a spray bottle some horticultural oil (organic) and blanketed my garden. Some burned leaves and a few sore muscles later, I now have an aphid-free garden thanks to my new weapon of choice!