Congress could investigate, or keep talking about it, or push for something to be done to negatively impact the parties that did it, or anything really. Even the slightest deviation from the present course of "don't say out loud that it happened and hope everyone forgets," would be a welcome show of some backbone.

>And how will that be different from trial by media?

Look at history to see the many things Congress can do when someone tries to pull something on them. They have options, they just aren't taking any.