How do you get dark, deeply penetrated stains out of hardwood flooring?

We have an elderly dog who often forgets to ask to go outside, and ends up just peeing on the floor wherever he is. One of his favorite spots to let loose was at the base of a plastic bin that we had setting on the floor. Mr. Laureth would clean up, but didn’t realize how much pee had gathered, lurking under the plastic bin until we moved it aside one day, and noticed the hardwood floor was blackening with rot, in exactly the same shape as the bin’s “footprint.”

We cleaned off the crusty ick with bleach-water, but the wood itself is nasty. Short of sanding the whole floor down and re-varnishing it, is there a way to simply spot-fix that area of the floor?

I may be hoping against hope here, but maybe someone here at Fluther is an expert in hardwood floor denastification.