Cheney wasn't in the best health, but the GOP would have a HUGE advantage due to a sympathy vote if Cheney ran and Cheney of all people would know that. Cheney would probably immediately pick a "strong" VP that he had confidence in to take the reins if he died in office. I don't know who exactly that would be. Cheney uses his control of the agenda for the next year to push through GWB's Social Security reform plan.
Unlike being pretty close as per OTL, the 2004 election is more or less a foregone conclusion from the get-go. The Democrats, deciding to go big or go home, nominate Howard Dean, hurting them on the campaign trail. Cheney wins the election easily(probably without having to get SoCons out to vote with anti-SSM referenda).
Katrina still happens, I doubt it'd turn out any different, and the President would take the blame. Cheney would probably be harsher on Pakistan than GWB was as well, and the Iraq War would inevitably grow unpopular and Cheney's approval ratings would sink with it(though I doubt he would go as far low as Bush's OTL). The Democratic victory of 2006 happens, to a lesser extent than OTL, the financial crisis and recession still happen, I doubt Cheney would've done any of the bailouts.
The Democratic primaries are a disastrous and indecisive slugging match between Hillary Clinton and Barack Obama at first, but voters eventually coalesce around compromise candidate John Kerry.
With Cheney refusing to participate in Republican politics(partially because he's a liability and partially because he doesn't feel like it) the GOP campaign goes a bit differently: Cheney's VP(whoever he or she is) has had their image dragged down by the Administration and decides against running. Social conservatives, dissatisfied after 8 years of being out in the wilderness, nominate Mike Huckabee, who turns off fiscal conservatives and rural non-evangelical whites, especially after a disastrous debate performance.
Kerry wins easily, though he fails to carry IN, NC, CO, or VA as Obama did OTL.
The Kerry administration proves disastrous, with Kerry seen as half-hearted, ineffectual, and waffling by both parties. Dissatisfaction with his handling of the recession leads to a good year for the GOP with ~230 seats(less than OTL, Kerry is unpopular, but no Kerrycare to rally around).
As things drag on, Kerry sinks further and is thrown out of office by the Jindal-Ryan ticket in 2012.
EDIT: Maps, maps, maps!
2004(I just added an extra 5-point GOP swing to OTL's 2004):
2008:
2012: