Cream generally results in a better texture than milk when making mashed potatoes, but assuming you have your reasons for using milk (health concerns, allergies, etc.), there is only one way to literally dry it out, and that is to evaporate the water. Toss it in a pot and simmer it uncovered until it reaches the consistency you want.
Because it's potatoes, you might have to stir it very often or even shake the pot around in order to prevent burning and a nasty mess. This is actually what I always do to potatoes before mashing them (after boiling) to dry them out. I've never attempted it after mashing them, but the potatoes are already cooked, so I doubt that you can do any more damage that way.
If you just want to thicken them then add some savoury ingredients. I really wouldn't recommend flour, potatoes have enough starch already, but I'll often add cheese to my mashed potatoes. Hard cheeses like parmesan are particularly good if your aim is to soak up some of the excess moisture.
I'd also suggest, in the future, that you whisk in your liquid gradually to avoid this exact problem.