Try pouring a lot of water on it, if that is feasible -- I know you said the ground is frozen, if its not too much work, take a gallon jug (or a couple) of water at about body temperature out there, and after they pee, pour about half a gallon on each spot to immediately dilute it out.
Also, sprinkle some sugar on each spot, about a quarter cup -- this feeds the microbes that will use up the urea, and it really does help. Unfortunately, if the ground is frozen, that puts all of them into suspended animation.
When spring comes, you can dig out the dead patches and backfill with clean soil and either plugs of sod or seed -- I always seem to have sod that grows over the edges of the walks and driveways, so I just go out there in April and edge, then use that to fill bald spots in the lawn. My major problem isn't my dog, but actually moles that make those big mole castles, leaving behind dead spots. Plus where I gouge things with the snowplow if I go over the edge -- but this winter, so far, I haven't had to plow at all, and barely have had to shovel.