I don't know about sauna action (I thought it'd be more about steam), why not try pebble trays? They help w/ humidity for plants that need it.

I don't think it would hurt anything, but I also don't think it would help much either. I've seen things that say that short term efforts to increase humidity like misting don't really make much of a difference to plants.
If you wanted to, you could consider setting up a grow light in the bathroom to allow some plants to become permanent residents in there. :)

Have you ever thought of growing them in your kitchen?

Next to the bathroom, the kitchen can be the most humid room even in winter as long as you cook with water very often. Just a thought:-)

Mike