For five years Mount Rainier National Park was my stomping grounds. It's a bit of a drive from Seattle, though. I also haven't been back since the big storms a couple of years ago. So, here goes...
There's a loop hike starting in Longmire that goes to Indian Henry's hunting grounds and back. It sort of circles Satulick Mountain. The last (or first) couple of miles is on the road, though. A true master might be able to do this hike in one very hard day. If you make good time it is easy to lengthen the hike over Emerald Ridge, but then you walk back on the Westside Road if you want a true loop. Or, you can just accept a little backtracking to Indian Henry's. Longmire has lots of parking to leave your car.
A non-loop hike that I like leaves your car in Longmire, then hitchhike to Reflection lakes, south of Paradise. (There actually might be a shuttle from Longmire to Paradise, but I'm not sure. Hitchhiking is easy.) Go up the Pinnacle Peak trail and turn west. Outstanding views, and you can tag seven non-technical peaks (Pinnacle, Plummer, Denman, Lane, Wampenayo, Chutla, and Eagle). Unless you try to tag Lane, stay out of the scree along that ridge- stay lower in the valley to make your way to Wanpenayo. Finding the easy way up Wampaneyo can be troublesome. Then, you will swear that you have to fly off a cliff to get from Wampaneyo to Chutla, but there is a very easy ramp hidden from view in some trees- look for it. It is also important to remember that the "trail" up Eagle peak is on the WEST side; do not try to climb the east side- it is technical. Supposedly this can all be done in one day, but I have failed to do so on two occasions (the second time was DEFINITELY my hiking partner's fault) and had to skip Eagle peak both times and go down the Longmire trail in the dark. So, a two-day hike might be more reasonable. It doesn't meet your mileage request, but these are HARD miles, and requires some orienteering on rough ground. You can skip bagging peaks to make better time, but you still have to go over Wampenayo to Chutla. Hmm. Hiking the Wonderland Trail from Longmire to Reflection Lakes would make it a true two day LOOP hike. There you go! Perfect!
A Rainier NP loop I've always wanted to do but haven't (DISCLAIMER!) starts and ends at Mowich Lake, circling the Mother Mountain ridge. On paper (maps and guidebooks) it looks very interesting.
I actually like the Sunrise area better than Paradise (less people), but those are mostly day-hikes in and out of the treeline. You could easily find some loop to do, though. Combining Sunrise, White River, and Burroughs mountain comes to mind. I'd go clockwise, though, or the hike up the mountain from White River to Sunrise would be horrible.
For the truly adventurous, climb 12,000-foot Mount Adams! The South Spur route is non-technical, requiring only crampons and an ice ax, no ropes. I've done it! In two days! You hike up to the Lunch Counter, camp there for the night, bag the peak the next morning and hike down. Actually, wear tough pants and you can participate in one of the longest glissades in North America, from the peak back to the Lunch Counter. Not a loop, though. And a damned long drive from Redmond. And that early in the season the road will still be partly snowed in, so you'd have an extra couple or three miles to hike. And you need to stop in Salmon River at the ranger station to get a permit. I've never heard of them running out of permits, though.
A great book is "75 Scrambles in Washington." I LOVE that book.