I had to go look for your other post to remind myself of the quilt you were talking about.
so you're saying that the pee caused the fabric to disintegrate? or is it just stained?
I use a product called "Kids N Pets" that actually does remove the stain and odor (so much that the critter that peed on the thing doesn't realize there's old pee on it). But you'd have to wash the quilt.
If you can replace the fabrics, you can probably rip the seams for the area that's been messed up, and replace with new fabric - it's just going to be a more difficult sewing job but it's not impossible. Ruth McDowell's "Piecing Workshop" provides instructions for how to sew an inset piece (which is what you need to do). It basically means doing a lot of pinning and going slowly.
Replacing the batting is pretty straightforward, but a pain in the butt. I don't know how you can fix the backing and make it look nice, unless you cut out an entire section and replace it whole.