Hi, bross. If you have a willow tree on your property, you should cut it down. They are dirty, dangerous to be around in a windstorm and actually illegal to plant in out city. One was here when I bought the house and my neighbors with swimming pools hated it.
Willow tree roots go down pretty far. I had to take mine out a couple of feet down. The stump was about 2 feet across and I first tried a chemical that was supposed to make the stump and roots rot away. The stump loved it and was growing nicely after 2 summers. And I had mushrooms growing along the roots for a couple of years as well.
I'm sure that there are chemicals that you can dump down the drain to try to kill the roots. I think copper sulfate will work for this but certainly do not try it without direction from somebody that knows more about it than I do. Buy a product especially for this and use it strictly according to package directions.