How to declog coffee maker clogged by milk
When you ran milk though the machine you cooked a lot of fat and protein into the components, which isn't good. The only "easy" thing I could recommend is to try to run some clear vinegar through it, the acidity may break down the milk residue and clear the machine. 

The only other way to get it working isn't as easy as it requires manually removing the residue using brushes. Inter-dental brushes are thin and bendable, you should be able to get them into those holes on the top picture. If you can get at the heating element with an old toothbrush that would be good too. Other than that the only thing I could think to try is a complete tear-down of the coffee machine, which may or may not be something within your skills or worth your time. Just remember to unplug it before you attempt any invasive repairs!