I'm not an expert on Michigan taxation.
Having said that, in general, most states income tax that the income being taxed is physically earned in the state. For example, professional sports; many states & cities tax player etc., on the pro rata income earned on a game played in their state/city...
In some instances where a person resides in one state and performs work in another state, unless there is a reciprocity between those states, a person may have to pay tax in both states.
Based on what you stated, you live and work in WA...and the work you're performing as an independent contractor is in WA. In my opinion, I don't think you would have to pay any tax to MI. No physical presence in MI. I don't know what the "staffing" company does, but you're not an employee of the staffing company, are you? By virtue of the being and "independent contractor" you're not an employee. Thus it would appear to me the "staffing company" is merely putting you in contact with someone who wants you to perform independent contractor work.
Again, my opinion only.
**Disclaimer: Effort has been made to offer correct information; but due to the discussion forum limitations, the poster disclaims any legal responsibility for the accuracy of the poster's response**