I am only licensed to practice law in the state of Ohio but have experience representing landlords and tenants regarding mold issues.
Based on that perspective and the facts as you explain them, I don't think that running a dehumidifier will solve your mold problem.
For significant mold growth to occur, there must be a source of water (which could be invisible humidity), a source of food, and a substrate capable of sustaining growth. Common building materials, such as plywood, drywall, furring strips, carpets, and carpet padding are food for molds. See http://en.wikipedia.org/wiki/Mold_growth,_assessment,_and_remediation
With that in mind, I suggest that you make a written demand on your landlord to undertake an appropriate investigation and corrective action. If he/she expresses reluctance based on cost, you may want to encourage the landlord to tender the matter to his/her insurance company. Tenant claims based on mold should be covered by the landlord's insurance policy. If the landlord still refuses to take any action, you may have to engage an attorney licensed in the state of Washington to file a legal action against the landlord.
At the very least, the source of water intrusion must be repaired first. After that is completed, replacement of the damaged wall, flooring and carpet should be completed.
In addition, the local health department's suggestion for having air samples taken is a good one. You should also consider testing the mold directly to determine which type of molds you are dealing with. Some are more harmful to human health than others. The cost for such testing could be in the $300 -$500 range depending on the number of samples taken. You should be able to find a qualified mold testing and abatement firm in your area through a web search.
Finally, you should also consider staying out of the contaminated room as much as possible until the remediation is completed.
I hope this is helpful. Good luck.