With all of my kids I had an area of the house where they could drink. . .

actually it was the dining room table. I got sick of finding disgusting sippy cups in odd places. So they sat at the dining room table, they drank and then they put their cup in the sink. Sippy cups only got water. So in your case, that's what I'd do. Stop filling the sippy cup with milk, and make a place where your toddler has to drink the milk.