The answer to your inquiry is just as you correctly surmise in the text of your question itself. The estate income tax return, of a (deceased) person whose legal domicile (residency) was the state of Illinois, produces Illinois-source income, which is taxable to the recipient(s) by Illinois.
In other words, the Schedule K-1-T you received is "telling" you that you do need to file an Illinois tax return, to report the income described therein.
Since you are residents of Washington state yourselves, this particular Illinois return will be a nonresident return. And, fortunately for you, since Washington has no income tax system of its own, you won't have to pay any Washington taxes on the Illinois Schedule K-1-T income. However, you do have to report the income on your federal tax return, as well as an Illinois nonresident return.
Moreover, you'll have to paper-file your Illinois nonresident return, because the Illinois Schedule K-1-T instructions indicate that taxpayers receiving this document must include a copy of it along with their Illinois tax return (for Illinois Department of Revenue "matching" purposes one would presume).
That said, seeking professional tax preparation assistance -- even if just for this one year with the Schedule K-1-T -- is an option for you to consider. Alternatively, if you feel up to the challenge, you can certainly do this yourself in TurboTax (the program is capable of handling this kind of income).
If proceeding in TurboTax, then you will need to tell the program in the Personal Information section, at the beginning of the program, that you have income from Illinois. You will then enter your Schedule K-1-T (for Illinois) in the federal side of the TurboTax program, just as you would with any other K-1.
To do this you can locate the Find / Search box in TurboTax, type in the exact search term "k-1" and click the button. Then you can click on the Jump To link that will appear directly beneath in the search results. This will take you to the main interview screen where you can input your Schedule K-1 data. Please see the screen-capture images directly beneath this text for a visual illustration; simply click the pictures to open.Then, after completing the rest of your federal tax return, you will need to be careful as you proceed through the Illinois nonresident tax return. You will be asked at some point in the Illinois state module about your Schedule K-1-T income. You will need to answer those questions appropriately, making sure that Illinois does not tax you on any of your other income, except for the Illinois-source Schedule K-1-T items.
Put differently, you just need to be watchful that only the "right" income items are being taxed by Illinois, so that you don't end up paying any more in Illinois state tax that you absolutely have to.
Once your federal and state tax return are completed, you should be ready to file them. While a federal tax return can still be e-filed while containing certain Schedule K-1 income information, as is explained above, your Illinois nonresident return will need to be paper-filed.