A WA state punch card is required in ONP, but there is no separate ONP fishing permit. The Hoh might well be out from glacial melt next week if the weatherman is right about the weather forecast, so it may not be your best bet.
If the Hoh is in, I'd fish it upstream of Hwy 101. Access is not a problem, especially once you reach the park several miles upstream of Hwy 101. Plus, your wife will enjoy seeing the Hoh Rain Forest, so domestic tranquility can be preserved.
Personally, I'd look at fishing one of the rivers closer to Forks (especially the one that flows virtually through town but I'd fish it a bit further west, or the one that Hwy 101 runs along on the way from Forks to Port Angeles) because they are rain fed and aren't subject to glacier melt. Access is not a problem on these either.