I assiduously avoid snow, so I'm not much help there either--although my guess is you could go to Snoqualmie, closer than Baker, and find trails for snowshoeing.
I second the Fremont recommendation--it's a fun and funky place to spend a day. If you're in the mood for the outdoors, try walking in the Arboretum, and there's a really lovely Japanese garden there as well. Feeling arty, and Pioneer Square, south of Pike Place Market, is full of galleries.
"My predominant feeling is one of gratitude. I have loved and been loved;I have been given much and I have given something in return...Above all, I have been a sentient being, a thinking animal, on this beautiful planet, and that in itself has been an enormous privilege and an adventure." O. Sacks