For $5 you can trade that old wooden paddle for a motorized mash paddle. A paint mixer is the perfect tool for mixing your mash and destroying dough balls.
Because it's plastic, it won't damage your kettle, false bottom, or fittings. I still try to avoid hitting anything but when I do it's no problem. I also recommend running the drill on a lower speed unless you want steaming hot grain flung into your eyes.
This thing mixes the mash more effectively than hand-mixing and does it in a fraction of the time. I highly recommend you try it out.