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Generally - you pay taxes based on your place of employment.
Thus - if your place of employment in Washington - your employer should be registered in Washington and pay employment taxes there.
In your situation your employer likely maintained the place of your employment in Ohio and payed all employment taxes in Ohio - that is why they refused to amend your W2 form.
Assuming that you will file your Ohio income tax return as non-resident. - mark that you was no-resident in the header and report the state of your residency - http://tax.ohio.gov/documents/forms/ohio_individual/individual/2007/PIT_IT1040.pdf
On the schedule D line 63 (see page 3) - Enter the portion of Ohio adjusted gross income (line 3) that was not earned or received in Ohio.
You may need to add the form IT 2023 - http://tax.ohio.gov/documents/forms/ohio_individual/individual/2007/PIT_IT2023.pdf - see also instructions - http://tax.ohio.gov/documents/forms/ohio_individual/individual/2007/PIT_IT2023_Instructions.pdf
However as your company reported on W2 that all your income was earned in Ohio - you may expect that Ohio DOR will disagree with your tax assessments and will ask to provide evidence that your income was actually earned outside the Ohio. You may ask your employer to write a letter to the Ohio DOR as a proof that your actual work place was indeed in Washington.
In additional you may provide the Affidavit of non-Ohio Domicile for Taxable Year 2007 - Nonmilitary.
The income would be consider as from Ohio sources - if that income earned while you physically worked in Ohio - that is correct.
However - you need to confirm that - the most common accepted confirmation is a letter from your employer. But in your situation I doubt that your employer would provide such confirmation because they are likely not registered to do business in Washington and may not have employees in Washington.