WASHINGTON — Two months into full Republican control of Congress, GOP leaders are struggling to demonstrate they’re really 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 demonstrated anew his need to rely on Democratic help at key 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, for his part, had already bowed to the demands of Democrats and stripped contentious provisions rolling back Obama’s immigration policies from his chamber’s version of the 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 in which Republicans won control of the Senate and increased their House majority to the largest in some 70 years.

Two months into the new Congress, the severe limits to their powers are confronting both Boehner 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.