Rush Limbaugh is not the leader of the Republican Party. That's because they don't really have one.
True, by definition, there has to be one Republican who has more influence than any other on the medium-to-long term direction of the party. Maybe it's Rush. Maybe it's Sarah Palin. Maybe it's Newt Gingirch, Charlie Crist, Mitch McConnell, John Boehner, Sean Hannity, John McCain, Mike Huckabee, Michael Steele, George W. Bush, Arlen Specter, Mitt Romney, or T. Boone Pickens. I don't know. I don't know that I care.
But that's not usually what we think of when we think of a "leader". Instead, we think of a "leader" as some individual who (i) almost everyone else in the party defers to, and (ii) has the moral and/or institutional standing to speak on behalf of the party as a whole. There is no guarantee that any one individual will satisfy these criteria at any given time. And in the Republican party at the moment, pretty clearly nobody does.
Rush and Michael Steele, for instance, went after one another in the most public way possible. Leaders do not, as a matter of course, have other major figures in their party seeking to humiliate them. They are too intimidating for that. Nor, for the most part, do they seek to publicly humiliate others. Their power is understood, and does not require such displays of force (although what happens behind closed doors might be a different matter).
Indeed, when a party doesn't occupy the White House, and hasn't designated a Presidential nominee, it usually won't have a "leader". Every now and then you'll get a Gingrich circa 1994 type, but these situations are exceptional.
Opposition parties, in fact, can sometimes function fairly effectively without a leader. Who was the leader of the Democratic Party from, say, 2005-2007? Howard Dean? Nancy Pelosi? Al Gore? Bill Clinton? Hillary Clinton? Harry Reid? Markos Moulitsas? It's not at all clear (although Dean probably has the strongest case). But these folks generally worked well in tandem with one another, and the party did quite well for itself. Everyone was working from the same playbook, which was to parlay George W. Bush's unpopularity into huge electoral gains.
The Republicans' problem is not that they don't have a leader, but that they don't have a direction. The House Republicans aren't on the same page as the Senate Republicans, who aren't on the same page as the Red State Governors, who aren't on the same page as the Purple State Governors, who aren't on the same page as the Beltway Elites, who aren't on the same page as the Media Elites. Nobody is happy with their statition in life. Nobody, except maybe Newt Gingrich and John Huntsman, is in anything other than survival mode. Nobody (including Steele, the RNC Chairman) is willing to put the best interests of their party ahead of their desire to mark their territory and increase their personal sphere of influence.
This is not the sign of a healthy party -- but Rush Limbaugh is the least of the Republicans' problems.