Because the eviction process and the family factor, make this a delicate case. You will need a lawyer in Seattle. The fact the there is no rental agreement in place you will start with unlawful detainer and get an order to show cause, because the order must be signed by the court prior to service the timelines and dates for hearing are critical. That is why a lawyer who does evictions on a regular basis is key for you.
One missed step in eviction can cost you more than a lawyer for the entire process.
From the information provided, your best course of action is to be very kind, tell your brother's girlfriend what you plan to do. If she does not want to sign a rental agreement and temporarily rent from you, ask her nicely to leave, and then do as much as you can to help her relocate. She may seem like a guest, but because she is living there she has legally recognizable rights to remain under certain conditions - even those conditions that don't make sense if she's not paying anything to be there.
Your alternative is to resort to a legal system since your brother's girlfriend has legally recognizable rights in living in your home. The action you'll need to pursue is called "unlawful detainer." It is expensive, emotionally and financially draining, and takes a long time to evict the "resident/tenant." There are also many critical rules that must be followed to properly effect the eviction, or you the process must begin again. If you haven't been through this process before, I recommend hiring an experienced attorney to help you if it comes to that.
If your brother's girlfriend is damaging your home, she will be required to repair the damage. Unfortunately, it will likely require a legal action (lawsuit) to get her to do that. An alternative there is to work with her to move out quickly and offer to repair the damage at your own expense ... in other words, "waive" your right to pursue her to pay for the damages in return.
Good luck. Mark Arend