King left a meeting with Boehner and other Republicans without the anger that had led him to rip into the speaker on the House floor, calling Boehner's attitude "dismissive" and "cavalier."
"It was a very positive meeting," King said afterward.
King told NBC's "Today" on Thursday that Boehner had ultimately come through for New York and New Jersey and said he expected Boehner to be reelected as speaker. King said he would vote for Boehner.
Boehner, R-Ohio, assured the lawmakers that the money from the two House votes would roughly equal the $60 billion package of aid that passed the Senate, King said.
Since the votes will be taken in the new Congress, the Senate also will have to approve the legislation. If the House, as expected, approves the $9 billion flood insurance proposal, the Senate plans to move quickly in hopes of approving the aid on a voice vote Friday. The flood insurance money will help pay for claims by home and business owners with coverage.
Sandy was the most costly natural disaster since Hurricane Katrina in 2005 and one of the worst storms ever in the Northeast.
Boehner's decision Tuesday night to cancel a vote on Sandy aid before Congress ended its current session provoked a firestorm of criticism from New York, New Jersey and adjacent states where the money will go.
According to King, Boehner explained that after the contentious vote to avoid major tax increases and spending cuts called the "fiscal cliff," the House speaker didn't think it was the right time to schedule the vote before the current Congress ended.
"What's done is done," King said. "The end result will be New York, New Jersey and Connecticut will receive the funding they deserve."
The Senate approved a $60.4 billion measure last Friday to help with recovery from the October storm. The House Appropriations Committee has drafted a smaller, $27 billion measure for immediate recovery needs and a second amendment for $33 billion to meet longer-term needs.
The $9 billion in flood insurance money to be voted on Friday was originally in the $27 billion measure. The votes on Jan. 15 will be for $18 billion in immediate assistance and $33 billion for longer-term projects, including projects to protect against future storms, King said.
Much of the money in the proposals is for immediate help for victims and other recovery and rebuilding efforts.
Some $5.4 million is for the Federal Emergency Management Agency's disaster relief fund; $5.4 billion is to help transit agencies in New York and New Jersey rebuild; and $3.9 billion is for the Housing and Urban Development Department's development fund to repair hospitals, utilities and small businesses.
New Jersey Gov. Chris Christie, a Republican, was among those sharply criticizing Boehner before the speaker changed course, complaining about the "toxic internal politics" of the House majority.
King was part of an angry chorus of New York and New Jersey lawmakers from both parties who blasted Boehner, with some saying his move was a betrayal. He told Fox News on Wednesday: "These Republicans have no problem finding New York when they're out raising millions of dollars. ... I'm saying, right now, anyone from New York or New Jersey who contributes one penny to congressional Republicans is out of their minds," he said.