As someone pointed out, we have the first internet-troll President. He's like the troll who shows up on your favorite message board and says that the thing you like is stupid. The media are the 20 message board regulars who spend hours writing long missives trying to prove him wrong when they should simply ignore him. Everyone knows you don't feed the trolls, because that's exactly what they want.
They should focus exclusively on legislation he's pushing for or getting passed, and ignore his Twitter feed. And like you say, focus on the rest of the administration and Congress and such. They do cover that a bit but it does get lost in all the fluff. Which is the point of the fluff, I'm sure.
While a good analogy to a point, internet trolls do not have much of an impact on anybody unless you let them. Being the president gives a person pretty decent power over a ton of people. Not sure if ignoring the problem makes it any better.