For a bit of background WaPo relates how the request was made. It sorta came from Sund, the Capitol Police chief (who has resigned since then) but oddly only after someone (unnamed) DC official suggested it to him. An the Pentagon was initially reluctant to approve it. Note that this refers to sending the guard to the Capitol itself (where Sund had command), not to surrounding DC areas:
The Capitol Police, the law enforcement force that reports to Congress and protects the House and Senate, hadn’t requested help from the Guard ahead of Wednesday’s events. But early Wednesday afternoon, its chief made an urgent plea for backup from 200 troops during a call with top Pentagon and city officials, according to officials familiar with the call.
On the call, Capitol Police Chief Steven A. Sund was asked whether he wanted help from the National Guard. “There was a pause,” one of the D.C. officials said. And Sund said yes. “Then there was another pause, and an official from the [office of the] secretary of the Army said that wasn’t going to be possible.”
Mayor Muriel E. Bowser (D) confirmed that account in an interview with The Washington Post, saying Capitol Police “made it perfectly clear that they needed extraordinary help, including the National Guard. There was some concern from the Army of what it would look like to have armed military personnel on the grounds of the Capitol.” One concern was whether the Army had been invited by Congress.
A U.S. defense official said the Army general on the call didn’t formally deny the request but rather reinforced the negative optics of having uniformed personnel inside the Capitol, a point on which Bowser had agreed, and later checked with the chain of command. The defense official said Bowser agreed that if further support was necessary, D.C. police would provide it inside the Capitol, and the Guard would backfill D.C. police positions away from the building.
The defense official said the military wanted to be the force of last resort, and that military officials had urged Bowser to request more support from federal law enforcement but that she didn’t do so until Wednesday.
Higher-up leaders at the Pentagon then evaluated the request and activated the full D.C. Guard, in addition to later calling the governors of other states to send their Guard forces as reinforcements. The officials also lifted limits on the Guard for the new mission, arming guardsmen with riot gear, but not guns, before they headed to create a perimeter around the Capitol.
Alas the WaPo story doesn't detail how the decisions in the Pentagon were made (although the accepted answer deals with that.)
However the story above may explain why the Army discussed the matter with Pence (formally the Senate President) and the other leadership of the two Houses, as they were seemingly very reluctant to appear to go into Congress only at the request of a police chief.