Mike Pence for president, I say — president of the United States Senate. It’s just the assignment for a pickle like this, when President Trump isn’t on speaking terms with his own party’s leader in the upper house.

Mr. Trump is understandably upset at the failure of the Senate to pass health-care and tax reforms. It’s gotten so bad with Mitch McConnell, according to CNN, that the two Republican titans have been cussing each other.

Privately, the New York Times claims, Mr. McConnell has even been voicing doubts that the Trump presidency can be salvaged (not that the majority leader’s own approval rating in Kentucky is, at 18%, so hot).

In Phoenix Tuesday, Mr. Trump mercilessly mocked the state’s senior senator, John McCain, a lion of the president’s own Republican Party. That’s because Mr. McCain cast the deciding vote against ObamaCare repeal.

Mr. Trump also tore into Arizona’s other senator, Jeff Flake, another Republican. Plus, Mr. Trump has been feuding with the GOP’s Lindsey Graham of South Carolina.

One might think that a president who is facing at least the possibility of impeachment would be more solicitous of such senators. If Mr. Trump were impeached, remember, the Senate would be the jury.

That’s not the Trump way, though. Heaven forfend. And this is where the tough but affable vice president ought to be able to help — and, as a bonus, make a long overdue constitutional correction.

One can even argue that, technically, the vice president isn’t even part of the executive branch. Unless the president quits, is ousted or dies, the veep’s only enumerated constitutional duty is to preside over the Senate.

So why does Mr. Pence need two offices in the White House — or even one? Why does he play the prop to Mr. Trump, introducing him at political rallies like he were a valet?

The point is not to distance Mr. Pence from Mr. Trump. Rather, to position him so he could play the most effective — and constitutional — role in helping the Trump presidency.

Mr. Pence could move his base out of the White House and Executive Office Building. He could set himself up in the Capitol, where the “vice president’s room” is one of America’s most magnificent offices.

The vice president doesn’t need anyone’s permission to do this — though realistically he would certainly need Trump’s assent for such a tack to be successful.

So why doesn’t Mr. Pence re-base himself on the Hill and get to work shoring up support for the president’s program? And restoring the vice presidency to its original and sole constitutional duty.

The plain language of the Constitution suggests the vice president doesn’t have much choice. It says he “shall” be president of the Senate. He may get to vote only when there’s a tie, but can preside when he wants. (It’s only when he’s absent that the Senate gets to pick a temporary president.)

What’s so attractive about Mr. Pence moving to the Senate is that he’s perfect for the job. He’s a policy and principles person who’s four-square for the GOP platform. And he’s got experience on the Hill — he was a six-term congressman before becoming governor of Indiana, and in his last term he served as chair of the Republican Caucus, a leadership position.

True, the Senate isn’t the House — but legislative experience and a history of working with Hill colleagues should be much in demand in the Trump administration.

It’s not just health care and taxes. In Phoenix, Trump talked about the importance of the Second Amendment. Yet national gun-permit reciprocity is stalled in the Senate. The president will also need Congress on board if military action against North Korea becomes necessary.

The Senate’s floundering on ObamaCare repeal shocked the 30 states that voted to make Mr. Trump president and Mr. Pence vice president. If the Republican senators can’t get it done, why not try the vice president?

No doubt the vice presidency has been much derided. John Nance Garner said it wasn’t worth “a bucket of warm spit” (or some other liquid). Vice President Hannibal Hamlin took work as a cook for the Coast Guard.

Maybe, though, Mr. Pence could make something of it. It wasn’t until Richard Nixon became vice president that veeps insinuated themselves into the executive branch in the first place. Mr. Pence could lead the counter-revolution.

The Hoosier could give a daily press briefing in his majestic office in the Capitol. He could try presiding vocally from the chair and working the corridors when he wasn’t presiding.

Mr. Trump could certainly use the help. It might not work. But it’s hard to see how it could hurt. It’s not as if the current leadership of the Senate has got a lot to show for itself.

A version of this column originally appeared in the New York Post.