The dog, your dog, was the source of the stain. You could ask for pictures, the area should be fairly dry by now. If there is even the smallest stain or discolor, you could graciously pick up the cost for the carpet clean, it shouldn't be that much for one room.
Look at the situation from her viewpoint, she was trying to be a good Samaritan by letting you in and letting you get help in sub zero weather, and she got her carpet peed on.
For some dog owners, that is not a big deal, but if it is a big deal to her, why not just remain good neighbors and pay for the clean?

I agree with Lucille- you have to pay for the clean and I think give a big apology for the incident. She kept you and doggy safe and warm and the carpet clean should not really be that big of a deal. Poor Fido may have had frostbitten paws if she would not have come to your assistance.

Pay for the cleaning please. If one peed on my carpet I'd be upset and probably want to replace my whole carpet. I'd even be thinking of the urine in the subfloor. I'm probably obsessive or compulsive but also just not a pet person.
Regardless, if there wasn't a spot there before and she says there is one now, it's because of the dog marking it's spot and/or subsequent actions taken.

You call the carpet cleaner, though, and just have him do a spot clean just the spot. You're certainly not responsible for cleaning her entire carpet.

Ask her when the best time for her to have the cleaner come in is, and tell her you'll have your cleaning guy come then to get THE STAIN out.

Nope - if whatever he uses to clean just that spot changes the appearance of her carpet on only that spot, she may still be worse off than she was before she allowed you and the dog into her home. You are responsible for making it right. Unless her carpet is otherwise completely filthy, you may have to pay to clean the whole carpet. Next time watch your dog more closely - expensive lesson learned.

What should I do? Isn't it obvious? Sorry but it is to me. Not all persons are pet people, you may have met one. Just pay to have her carpet cleaned and I would be grateful no one got bit. I can hear Judge Judy now.

If you have difficulties seeing the situation from your neighbor's viewpoint, lets look at it from your viewpoint:

If you think that lightning NEVER strikes twice in the same place, think again. Fast forward to next winter, it's below zero, you've done it again and you are locked out.

You ring the neighbor's doorbell.

A) You were gracious and generous, picked up the cost of the clean and sent her a box of pastries with a grateful note.

She let's you in, and asks that Fido stay in the kitchen with linoleum floors, and makes hot chocolate for you and gives Fido a piece of cheese.

B) You gave her a hard time for being a Good Samaritan, tried to do as little as you could get away with on the pee situation, acted like it was her fault that her carpet got peed on by your dog.

She peeps through the window and sees it is you. Doesn't answer the door. You finally find a more distant neighbor but not before you've ruined your new $200 boots and Fido got frostbite, which turns into a $300 vet bill.