That's the wrong question. Most of the problems with NASA can be traced to confused thinking at the political level.

The agency is about as well-run as such an agency can be (a mix of operations work and basic research, combined with a coordination and procurement role in the broader space sector). Cherry-picked stories like the mess around the Columbia crash make the agency look really bad, and there are some very vocal critics who allege very severe mismanagement (including a couple of ex-employees who've gone very public with their allegations of big conspiracies and mismanagement).

Most of the allegations are a case of the politicians shifting blame away from themselves

Remember, NASA is a government agency, with political appointees at the top, and congressional committees driving high-level agendas. Not to mention presidents who don't really get space technology, but love it for its rhetorical potential (ranging from "back to the Moon, and then On To Mars" to "Sputnik Moment!").

NASA has a very ambiguous multi-role mission: basic "humanity elevating" science, a more practical shuttling-stuff-to-orbit mission that competes with the decidedly commercial ESA and the commercial space launch capabilities of players like Boeing, and finally, a defense mission because there are times when the American military gets out of its depth in doing space stuff (the Air Force has its own space capabilities).

Look at the wildly changing political missions that NASA has had to deal with. A private-sector CEO would throw up his hands and give up on day one.

1. Compete with the soviets (50s)
2. Get to the Moon, explore the solar system and help build up the nation's engineering foundations (60s and 70s)
3. Star Wars, the Shuttle and Big Space ideas like Hubble (1980s, early 90s)
4. A decade of somewhat misguided cheap-and-quick $300 million missions that got support primarily as a backlash against Hubble level missions
5. A crazed moment of "return to the Moon" obsessions (this happened around the time I was entering the job market after my PhD in 2003. I interviewed at JPL which was at the time going nuts/schizophrenic trying to deal with the competing political pressures to prepare for a Back to the Moon pressure as well as finishing the job on the cheapie missions phase)
6. The Chinese Scare in recent years, as China launched the first Taikonaut into space

Much of the insanity is due to this constant jerking around at the political level. IMO, this happens because there is just so much money in a too-illegible system. Pigs at the trough. The solution is to split up NASA in ways that add transparency to what goes on, and prevents the money-grubbing from messing with broader intent.

1. Break off a coordination arm that functions more like the FAA: a policing authority without much to do with revenue flows. Keep the core mission-control and launch infrastructure with this body. They charge rent, launch fees, licensing fees, audit for safety etc.
2. Leave commercial space launch services ENTIRELY to the private sector and competing government agencies. This is where most of the money is, so let the market police it through competition.
3. Give the basic research mission to the NSF. They stop at the level of simple experimental satellites. Beyond that, the reduced NASA takes over.
4. Break up the remaining assets along manned/unmanned lines, purely for missions that are about basic science/elevating humanity. So Goddard and Johnson would go to the manned program for example. JPL would go unmanned. It is already effectively set up this way and the different centers shadow-box for budgets in the dark anyway. Just make everything much clearer and more transparent, with separate budgets and management.
5. The military can stick to its role and borrow these civilian resources as needed.

Manned and unmanned are fundamentally different technology sets with very different cost structures, science needs and political motivations. We have no hope, in America, of converging conclusively on one or the other as the "only way." We need a minimum "capability maintenance" budget for each, and then Congress can fight every few years over where to place their bets. Given the long lead times of any kind of space research, it would be nice if you could rely on 10-year stability periods (we are not talking iPhone apps; even the simplest mission has at least a 5-year timeline to launch), but unfortunately this is a case when the tempo of American democracy doesn't work with the tempo of space research unless a president gets re-elected.

For the record, I personally believe manned spaceflight is simply meaningless to pursue now. Getting to the Moon was a fantastic achievement that showed just how extreme humans could get in their questing for new frontiers, but at some point, unless someone figures out hyperspace or warp-drive or some game changer of that variety, it is time to be humble and admit that taking on the universe (heck, even the solar system) with manned spaceflight is a case of biting off just a teeny bit more than we can chew. I'd pick a manned Mars base of no economic value over feeding the starving children in Africa any day, but unfortunately even that isn't realistically achievable right now.

Without a game-changer, we should next revisit manned spaceflight when the Sun is going red giant and we need to get off the planet.