Story highlights House Republicans are gushing over VP-elect Mike Pence, Donald Trump's emissary to Capitol Hill

"When someone that's respected comes in, you can just tell -- it's a walks-on-water type feeling there"

Washington (CNN) Republicans on Capitol Hill are still feeling out President-elect Donald Trump -- but they're warmly embracing his vice president, Mike Pence.

Pence held court in private meetings with GOP members of the House, and then the Senate, on Wednesday.

Those visits are set to become more permanent: Like Dick Cheney, the soon-to-be vice president will have office space in both the Senate -- where he technically presides as president and will cast tie-breaking votes -- and the House, too. He also plans to attend weekly policy lunches, positioning Pence as President-elect's chief fixer on Capitol Hill.

It's too early to tell what role Pence will play in setting policy -- or whether he will consistently speak for the Trump administration on policy matters in a world where Trump can change the agenda with one tweet. But Republicans were gushing about the man they see as Trump's good cop -- one who bridges conservative divides where Trump at times deepens them -- just 16 days from inauguration.

"I think it makes a unique situation -- somebody who was elected in the class of 2000; somebody who understands the speed at which the House works. To have somebody like that as an advocate in the White House is going to prove to be very beneficial to us in the days ahead," said Rep. Mark Walker of North Carolina, who chairs the conservative Republican Study Committee -- the same role Pence once held when he was a member of Congress.

Read More