The lawn ranger

Since we acquired a female puppy in early May our little lawn has become a complete wreck, mostly bare earth with just a few pathetic strands of grass here and there. They say you should teach the puppy to pee and poop in just one area but as our garden is so small, and is nearly all grass - or was! - that would impossible.
I tried the Dog Rocks, which you put in the drinking water, and they made absolutely no difference at all. I tried watering where she had peed but this meant I had to watch her like a hawk and keep rushing out with the watering can in my hand - hopelessly impractical. Next I decided to hose down the whole lawn morning and evening, in the hope of diluting the urine, but that didn't work either and the ground became much too wet.
Last weekend I was visiting a friend who also has a female dog and a tiny garden but to my amazement her lawn was unmarked by any unsightly patches. She told me she puts a teaspoonful of tomato ketchup in her dog's food and this apparently does the trick.
My friend also said she'd heard tomato juice was best of all but because she doesn't like it herself, and didn't want to throw the surplus away, she opted for ketchup.
Does anyone know if using either tomato juice or ketchup is OK for the dog's health? And if using the ketchup, should it be the low salt, low sugar version?
Julia Lewis