Legal Question in Immigration Law in Washington

In short, you can't. "Work permits" can only be obtained by those who are eligible for an immigration benefit (greencard, asylum, employment based nonimmigrant visa, etc). If you had a legal entry but overstayed and you are married to a U.S. citizen, then you may be eligible to apply for a greencard (depending on if you have any arrests, deportations, etc). You may want to speak with an immigration attorney to discuss your particular set of circumstances, since you are at risk for "removal " if you are out of status.