Two thoughts in response to Marty's provocative questions :
First, I think having a relatively independent DNI probably is manageable constitutionally. We already have, after all, an FBI director who is appointed for a term of years that does not coincide with that of the appointing president. While the DNI is higher up the food chain, I suspect the office could be structured so as to look pretty similar. The more formal one makes the DNI's independence—in other words, the harder his removal is for the president to effectuate—the more difficult the question becomes. But at a minimum, it should be possible to create an office with a term of years and a strong norm against removal for reasons other than misconduct.
All of which seems to me a perfectly dreadful idea, and I'm frankly a little bewildered by its attraction for the people most offended by the intelligence policies of the current administration. After all, having a long-term occupant of that office would ensure continuity across administrations in an area in which there is simply no political consensus as to the proper posture of the executive branch. If you imagine it existing now, it would allow Bush to appoint Obama's DNI. The reason the FBI director's term of years is defensible is that Americans basically agree on the apolitical nature of the investigative function and want to insulate it from the shifting political winds. A similar consensus, I suppose, exists for much intelligence collection and analysis. But no such consensus exists for a lot of intelligence policy over which the DNI has charge. Would we really want Bush to appoint Obama's point person on warrantless wiretapping, renditions, and interrogation?
An Obama administration would presumably handle a lot of things within the DNI's purview differently from a Bush administration. And the Republican who runs against Obama four years from now, should Obama win in the fall, would presumably criticize and promise to change Obama's intelligence policies. If we make the DNI's position apolitical, we greatly reduce the capacity for political debate over and change in intelligence policy.