Best wilderness soaking hot springs in a national park in the US are about 20 miles outside of Port Angeles up the Elwha river. But you're got to ride West on the 101 out of Port Angeles.
From Port Angeles or even Port Townsend, I'd recommend riders go west unless they have to be in Oregon in two days.
The ride to the west is unquestionably best.
The very senic 112 to Neah Bay can be tackled after the hot springs by riding the enjoyable Joyce Piedmont Road between the north end of Lake Crescent to the 112 route to Neah Bay. I'd recommend heading to Neah Bay (grocery store, but dry reservation), then the ride to Cape Flattery, the most western point in the continental US - short hike to that view. that's a point to tick off the list, btw.
camping on the beach just south of cape flattery, pay the family that runs it, or hide yourself in the woods up on the Cape. The roads inland from the Cape Flattery provide a short clic to an unsigned, but easy to find overlook (the locals all know about it, but don't ask for directions, just find it on your own) to camp at that has a view of the sunset over Cape Flattery- an amazing place to stop or camp if you've got the wherewithal, but the beach campground is legit and a good way to pay back the tribal members that have reign on that patch of america.
then backtrack to the 113, or ride thru the logging roads of the rain forest to check out Historic Lake Ozette and the sea stacks of the coast via a side hike to the beaches if unwilling to backtrack to get to the 113, Sappho, and then onto Forks, the only real city between Port Angeles and Ocean Shores.
Even though there's not much civilization out there, don't forget the park lodges and stores at Lake Crescent, Klalaloch, Lake Quinalt. A cyclotourist can, should spend a couple of hours. I'd suggest timing your visits to catch a breakfast at the dining rooms at each of these National Park lodges, before walking the grounds and short hikes to some of the biggest evergreens in the world.
the ride thru the west side, not the east, takes a rider thru verdant temperate rain forest found in only a couple of places in the continental US. The Olympic may be the only temperate rain forest in the continental US, I forget. (these are mostly found chiefly Canada's Vancouver island, the coast ranges north, and Hawaii's volcanos, but i digress) If you ride east, you miss this entirely.
The Ocean Crest at Moclips presents the next civilized stop, before a small grocery store in Moclips, then on to Ocean Shores that will seem like a seething beach resort town after the quiet of the rain forest.
A rider with a couple of extra days is well advised to take the short ride from Forks to the sleepy La Push, to hike to Second and Third beaches on the Olympic coast. Camp on second beach, or third beach, absolutely sublime. This is about 15 miles from Forks.
The real beauty on the Olympic peninsula is the side trips. Take the ride down from Hurricane Ridge. I wouldn't recommend a tourist hump their loaded bike to the top of Hurricane, but if you want stunningly beautiful vistas then a 5,000 foot descent looking out into the mountains of canada across the Straight of Juan de Fuca, finagle a ride in a pickup truck with an unloaded bike to the Hurricane Ridge visitor center in the Olympic National Park before leaving PA.
IF you still decide to ride the pedestrian east side of the 101 and can ride mild dirt roads, there is a wonderful shortcut up and over the olympics between PA and Quilcene, that rolls past the Dungeness Forks campground, to one of the best paved wilderness descents in Washington state. I've found it to be the best backroad descent in the state, but there may be some out towards Idaho i might have missed.
I can help with directions for that if you decide to ride the east side.
This is the only thru high road that cuts across the olympics btw, the other high roads on the entire peninsula are ALL either dead ends, or low.
The Quilcene crossover high road is one of the few attractions on the East side worth detouring for.
Like i said earlier, the West half of the Olympic Peninsula is clearly best.
The real trick for the bike tourist is to plan the time, and chart out a wandering route on the Olympic Peninsula, to avoid developing road blindness from sticking to the main road around.