You might be able to spray soapy water and wash it down into the soil. Activated charcoal or generic kitty litter (zeolite clay, not the baking soda stuff) will help absorb it.

Is Killex a spray or a granular? Spot spraying weeds with herbicide is about the safest way to do the chemical spray. The worst way is to apply a granular product over the entire lawn.

Dandelions can be easily removed from a standing up position with the Weed Hound tool. I have not looked for one recently but I got mine at Home Depot last year for $25. You hold the tool in your hand, aim it at the plant, step on it, twist, and remove it. After you get the hang of it, you can remove them as fast as you can step on them.

Its mostly 2,4-d as an active ingredient.

Think of your lawn as layers. you have soil, roots/soil and grass. The killex is in the roots/soil level. if you put down activated charcoal or carbon or kitty litter, this will be on/in the grass layer. It wont absorb the killex unless you till your yard and mix it in. Save the money and let time pass. In about a month or less the killex will be out of the soil.
2,4-d is proven and is safe for residential use. I would not worry.

You are basically trying to close the door after the horse has left the barn. Once you spray herbicide on the lawn and let it dry, there really isn't much you can do to remove it. The active ingredients have already entered the leaves and migrating throughout the plants. The herbicide that reached the soil is already starting to break down. It dissolves in water, but short of a flood, you aren't going to be able to flush enough water through the area to do much good. Even if you could, you would just be washing the chemicals onto someone else's property.

The only thing that really is going to remove it is time. The active ingredients decompose in the soil on their own. You can speed the process by making sure it stays relatively moist.