Okay, many of you helped me with the new dishwasher I got to replace
the old one. Thanks for the opinions I got. Now I have to deal with
what the old one left behind.

The adjacent cabinet is rotten from water damage and mold/mildew. I
never knew it was leaking until the day we saw the inside of the
cabinet. The floor of the cabinet is bad and must be replaced. I
can't tell if the cabinet wall is bad from the inside, but the outside
wall has some damage at the footer where it contacts the floor.

I know this has to be repaired, but I don't know what materials are
available to me. Also, I have a toddler running around the house, so
whatever I do needs to be done from start to finish inside of a day or
so. I don't want to take the countertop off, so hopefully there is a
way to get around that.

Any recommendations?

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No matter what happens someone will find a way to take it too seriously.