Since the website showed that 66% of the time it was the water inlet and 33% of the time it was the float switch, I decided to replace both, as the cost of the two parts together was less than it would have cost to have someone come out and diagnose the problem.
I ordered the parts around noon on a Friday and received them by 3:00 on Saturday, without paying for any kind of express delivery.
The videos on the website were perfect for showing me how to do the repairs, and the repair took less than an hour.