They made it to the 2nd round of the playoffs and brought the cup champs to an intense game 7. They are hammered with important player injuries and are on a small losing skid because of it. Coaching only can do so much, considering the circumstances it would be pretty unfair to Babcock at this moment to even consider this his fault and jeopardize his job due to reasons beyond his control.
Plus, I sound like a broken record here as I keep saying this, this team is still sitting 6th in the East and that's pretty damn impressive all things considered to be in a playoff position. Granted the Wings lead the league in OT loss points, but in this modern NHL points are points and that's what determines the overall standings.
There is one resolution here, it's not the coach, it's not the GM, it's the injuries. Get healthy, and they begin to play better.