The speed of the drill needs to be set to medium to fast which is about perfect for making mayonnaise. When mixing, you need three hands if you use a hand drill. If you are doing this alone, a drill press fitted with a wire wisp is very handy. Ordinary electric mixer blades may not produce acceptable results. You may be tempted to use a blender, but if you do, you will get a mess instead of mayonnaise.
Most blenders have only a small impeller at the bottom so they wont work for making mayonnaise.
Let the egg yolk mixture sit for 3-5 minutes. The risk of getting food poisoning from a raw egg is extremely low, but letting the lemon juice 'cook' it for a few minutes will make it even less of a risk. Remember that you are mixing a large amount of oil into a very small amount of lemon juice. This takes a lot of time. As you do this, it is very helpful to have a kitchen helper hold the quart jar for you or use a drill press if you are working alone.