Second the Death Valley recommendation - great winter hiking. Depending on when the snows come, Wildrose (9K') or Telescope (11K') peaks may very well be accessible without spikes. There are plenty of strenuous canyon hikes as well. DV is my favorite winter playground since I don't do snow/ice particularly well.
Olympic and Mt. Rainier National Parks are driving distance from Seattle, but count on a long day - 2-3 hours drive time each way. You can add Mt. St. Helens to that group as well. Plenty of good, hard hikes in these parks. The snowline is very, very low in coastal Washington, some years as low as 4000 feet elevation. If they're accessible, I'd recommend Hurricane Ridge in Olympic, and the Fremont or Rampart Ridge trails in Rainier.
Then there's always the Stairmaster . . .
