The ground does not sound very nice, Constance, what with dog mess and Jeys fluid! I don't think that I could face eating vegetables grown in that soil.
If it were my garden I would be diging it over thoroughly, adding lots of fresh topsoil and soil conditioner and enjoying a flower garden.

I agree, I wouldn't consider eating anything from it, especially root vegetables...urgh!

Stick to flowers as Bulbaholic says and perhaps get a few containers filled with a half loam, half compost mix to grow a few leaf veg. If you want to grow root veg, sow direct into the containers.

I have a dog and have trained her to use a piece of grass set aside just for her!! Why have you let your dog free to use the whole garden?. Sounds crazy to me. I agree with the 2 comments above. I wouldn't grow veg unless it is in a container. Plant with flowers and keep the dog to his part of the garden. Could grow a fence of some kind box, or lavander would smell nice.