Ok, here's a test to see if you need a helmet.

Stand in your driveway next to your bike. You can hold it or not. Doesn't matter.

Fall over.

Maybe you hit your head on something, maybe you don't.

Clearly it doesn't matter if you are going 5mph, 10mph, 20mph or simply standing still. Hitting your head can be just as debilitating from standing still, or at 20mph. Why take that risk?