WASHINGTON — Two months into full Republican control of Congress, GOP leaders are struggling to demonstrate they really are in charge.

The stunning House defeat Friday of a three-week spending bill for the Department of Homeland Security exposed Speaker John Boehner’s weakness in the face of rebellious conservatives.

It also again demonstrated his need to rely on Democrats at critical moments as the minority party’s agreement to a one-week spending bill helped the speaker get it over the finish line with only hours to spare before a threatened agency shutdown.

President Barack Obama signed the bill shortly before midnight.

Senate Majority Leader Mitch McConnell, R-Ky., already had bowed to Democrats’ demands and stripped the contentious provisions rolling back Obama’s immigration policies from his chamber’s version of the Homeland Security spending bill.

The two leaders face different, and often competing, challenges as they try to produce the responsible governance they promised voters after November elections, when Republicans won control of the Senate and increased their House majority to the largest in 70 years.

Two months into the new Congress, the severe limits to their powers are confronting Boehner, R-Ohio, and McConnell as they aim to chart a course for the final two years of Obama’s presidency. That path could help lead their party back into the White House, and perhaps even produce a few legislative achievements.

“Obviously we’re not getting good results, are we? I base everything on results,” said Rep. John Fleming of Louisiana, one of the many Republicans frustrated with the GOP’s performance, particularly on the Homeland Security bill.

“Our leadership set the stage for this,” Fleming said. “Yet we didn’t really see much messaging, coordination or communication.” Even though Boehner has large numbers on his side, it’s still not enough to ensure he can line up a majority on any given bill, especially on issues such as immigration, education or abortion. The GOP advantage is 245-188 with two vacancies.

McConnell is contending with Senate rules that give important rights to the minority party, which Democratic leader Harry Reid of Nevada is adept at exploiting. Republicans hold 54 seats, but that is six short of the 60 needed to ensure passage of most legislation.

Democrats united against and ultimately blocked GOP attempts to use the Homeland Security spending bill as the measure to overturn Obama’s immigration directives.

Many Republicans campaigned last fall on promises to stop Obama on immigration, and their inability to do so is infuriating to conservatives.