There should be some kind of enforcement possible separately for House, Senate, and Judiciary. It would take some thought how to properly arrange that, but it's foolish to have an obvious crime committed and no way to pursue it.
I've read suggestions that the House (and Senate) Sergeant at Arms might well have police powers beyond the immediate vicinity of the House, based on it's history in English common law, a history that extends back to Rome. I've been told that in the face of Justices refusal to enforce the law, the SAA could enforce the contempt citation against Lois Lerner, or our esteemed Attorney General. Don't know if the House has a jail, The likelihood of the House testing this is zero. But it is possible.
Not going to happen, unless the House recruits it's Sergeant at Arms from the military. With legitimate enforcement power. Not a court in the nation you can treat with contempt without looking at the world through bars.
There is. It's called impeachment in the House, and conviction in the Senate with the Chief Justice presiding.
Now, on the matter of the will to do that, others at FR has suggested that we have moved away from a three co-equal branches system to a two-party system. This means that checks and balances between the branches no longer work, because party loyalty decides who is guilty and who is not at any given moment.
What is criminal today is not criminal tomorrow, depending on whose party is in control. The three branches are no longer relevant.
-PJ