(CNN) Limited Sandy relief vote to occur on Friday

(CNN) -- After a firestorm of criticism from Northeast Republicans, the House of Representatives is expected to take up an aid package on Friday meant to address those still reeling in the aftermath of Superstorm Sandy.

Lawmakers will consider $9 billion in immediate assistance for flood insurance and will weigh another $51 billion in broader aid on January 15.

"We're getting what New York and New Jersey need, and that's all that counts," Rep. Peter King, R-New York, told reporters after emerging from a 20-minute meeting with House Speaker John Boehner and House Majority Leader Eric Cantor. "We're all big boys. We understand that all that counts is the bottom line."

That point was underscored Wednesday, when House leaders met with irate representatives from New York and New Jersey who felt they had been ignored by Boehner when he scrapped a planned vote late Tuesday on the full aid package.

The Senate, which had already approved the larger Sandy plan that the House refused to consider, is expected to sign off on the scaled-back version on Friday as well, according to a Democratic leadership aide.

Senators will hold off on any further action, however.

"On the second tranche, we will need to see more details before we decide how to proceed," the aide said. "As the Senate has shown by passing our bipartisan bill, we consider getting aid to the victims of Sandy a superlative priority, but we need to know more about the contents of the bill before deciding on a path forward."

Democrats were less mollified.

"It's really unbelievable how Speaker Boehner and his party could just walk away," said Christine Quinn, speaker of the New York City Council. "To promise us a vote weeks from now? Why should we believe him at all? It's just shocking."

In a statement, Boehner and Cantor said "critical aid" to storm victims should be the first priority of the new Congress, which convened on Thursday.

The comity contrasted sharply with outrage that had exploded earlier in the day over congressional inaction on the package, pitting even fellow Republicans against Boehner.

It was "disappointing and disgusting to watch," said New Jersey Gov. Chris Christie, blaming "the toxic internal politics of the House majority."

"New Jersey deserves better than the duplicity we saw on display," he said, adding, "shame on Congress."

Christie, a Republican, said he had tried to reach Boehner on Tuesday night after a vote on the aid bill was shelved.

"He did not take my calls," Christie said.

At a news conference, Christie said he joined people of his state in feeling "betrayed" and added that the move summarizes "why the American people hate Congress."

In a statement, Christie and New York Gov. Andrew Cuomo wrote: "This failure to come to the aid of Americans following a severe and devastating natural disaster is unprecedented. The fact that days continue to go by while people suffer, families are out of their homes, and men and women remain jobless and struggling during these harsh winter months is a dereliction of duty."

GOP leadership sources said Boehner was worried that it would be a bad political move for him to allow a vote on new federal spending after House Republicans spent the first part of the week criticizing him for not wringing substantial spending cuts in the fiscal cliff bill.

Additional pressure on Boehner came from Republicans who criticized proposed congressional "pork" spending in the bill that was unrelated to Sandy needs.

Civility was eventually restored.

"As far as I'm concerned, that was a lifetime ago," King said. "The bottom line is we're going forward getting what we believe is necessary."

King previously had slammed his own party. "The Republican Party has said it's the party of 'family values.' Last night, it turned its back on the most essential value of all, and that's to provide food, shelter, clothing and relief for people who have been hit by a natural disaster," King told CNN.