When a new Congress begins, both chambers have to begin from scratch with legislation, so the Senate's passage of a previous bill will be moot.
Michael Steel, Boehner's spokesman, said the speaker is "committed to getting this bill passed this month."
Before the House adjourned Wednesday, President Barack Obama urged a vote.
"It has only been two months since Hurricane Sandy devastated communities across New York, New Jersey, and Connecticut as well as other Eastern states. Our citizens are still trying to put their lives back together," Obama said in a statement.
"When tragedy strikes, Americans come together to support those in need. I urge Republicans in the House of Representatives to do the same, bring this important request to a vote today, and pass it without delay for our fellow Americans," Obama said.
Some of the harshest comments came from King.
Scott Mandel, vice president of New York's Long Beach City Council, told CNN, "The money was needed yesterday, and the fact that there's an obstacle in the way for whatever reason and a vote wasn't allowed to go forward was inexcusable."
The money would improve the city's ability to withstand damage from winter storms, Mandel said.
Fiscal cliff battle held up the measure
The tumultuous process of getting the fiscal cliff deal passed in the House had held up the relief measure, and many House Republicans opposed the size of the Senate bill.
"Leadership was all-consumed with the cliff procedure," Rogers told reporters off the House floor late Tuesday. "And they really have not had the time to devote to this because of that."
Sandy killed at least 113 people in the United States and left millions of people without power after running up the East Coast in late October. The storm hit hardest in New York and New Jersey.
Cuomo has put storm-related costs at $41.9 billion, while Christie has estimated a price tag of $36.8 billion.
The bill includes grant funding for owners of homes and businesses, as well as funding for public improvement projects on the electrical grid, hospitals and transit systems to prevent damage from future storms.
John Stone, a resident of New York's Staten Island, owned two homes before the storm. One was destroyed; the other was so severely flooded that it remains unlivable.
But he expressed no anger over the House's decision. "They'll just have to do it all over again, I suppose. What can you say?"
"It's a lot of money," he said, adding "there's a lot of other things they've got to do."
He tends to vote Republican and doesn't plan to turn away from the party, he said, although, he added, "I don't give them much money anyway."
He's been living with relatives in New Jersey.