The last step is to dry it out. Let the sheets hand dry, if possible, but keep them out of direct sunlight and away from heating vents because that is just like putting them in a hot dryer. It is especially important to keep dark colored silk sheets out of the sunlight because it will visibly fade them in a very short amount of time. But, because finding a safe place to air dry large silk blankets, such as king size sheets, can be difficult, some people have had good luck drying their sheets using the lowest temperature setting on their dryer, the one that is just barely higher than “air dry.”
If you think the sheets need ironing, you can use a cool iron on them while they are still slightly damp, making sure to iron on the reverse side (the non-shiny side). But I will tell you I have never found it necessary to iron silk sheets. If they are gently washed and dried (translation: not twisted and wrung and beat up), they don’t need to be ironed.
One other note about washing luxurious silk bedding – do not put anything else in the washer with them. Wash them separately from your other laundry.