It's an intractable problem. The issue of the debt ceiling stems from Republican frustration at the majoritarian (authoritarian) way the Democrats have run the government, just ramming through all of their ill-advised entitlement programs without a thought for how to pay for them and over the legitimate objections of fiscal conservatives. The Republicans, naturally, oppose increasing taxes to pay for these programs with which they do not agree (reflecting the opinions of approximately half of Americans), so the programs are inevitably funded through debt. Refusing to raise the debt ceiling has been an indirect way to try and strangle these programs, but the left-wing media generally collaborates with the Democrats to blame the issue on Republicans, and the Republicans inevitably back down.
The only long-term solution to our debt problem is entitlement reform (i.e. downsizing the entitlement programs and making individuals more responsible for the costs of their own healthcare and retirement). Since Democrats have an in-built advantage in the political system (giving citizenship to illegal aliens, bribing voters with handouts, etc.), it is unlikely Republicans will ever be able to control both the Presidency and the Congress with sufficient majorities to enact these reforms. Until reform happens, debt will continue to spiral out of control, and eventually the bond markets (and China) will no longer trust our ability to repay the debt. Interest rates will skyrocket, a financial crisis will occur, and then we will have no choice but to reform.
Churchill once said that Americans will always do the right thing, only after they've tried everything else. His axiom holds in this case as well.