Ned was prepared to do all that, but as always, succumbed to his fatal flaw and let his love sway his duty. Do you think it's an accident that Jon gets the lesson about the difficulty of choosing between love and duty so close to Ned's fatal choice? Sometimes doing what's right means accepting danger to those you love, which is why the Kingsguard and Nights Watch are supposed to put aside their families and not have wives or children. When Varys gave out the bullshit threat to Sansa, Ned should have laughed in his face - not only was the threat empty (as it would tarnish the Lannisters' reputation and render them untrustworthy in the eyes of those with whom they would deal in the future, and leave them with no further bargaining chips), but even if it meant Sansa would die, that was too damn bad. Is Sansa's life worth having not only a usurper, but an immoral and reprehensible ruler, on the throne of the Seven Kingdoms? Had Ned simply stuck to his guns a little longer, he would have lived, and the kingdom spared much of the strife that has gone on since.

As long as he was in the dungeons, he was under the authority of the lawful rulers on the council, who would never do anything so rash as his premature execution. Only Littlefinger sought his death, and that one later notes the importance of keeping his hands clean. Pycelle and Cersei had to be aware of the value to House Lannister's interests of keeping him as a hostage or removing him from the board by taking the Black.

His reputation and experience made him seem far more dangerous to them than Robb, whose military prowess was not yet revealed. Ned could have assembled a coalition of their enemies, and would have been trusted. He might have been the only one capable of reconciling the Baratheon brothers. Next to the proven martial ability of the heroes of Robert's Rebellion & Greyjoy's Rebellion, and their moral rectitude, Renly's style and charm would have counted little, and with Sunspear seeking vengeance against the murderers of Elia & her children, Ned Stark, the man who condemned Tywin Lannister as a murderer to Robert's face, would have been a natural ally. Had Ned taken the black, he could not have done any of this, and his House could not have sought revenge, which is why Varys and Cersei were willing to offer it.

But if Ned had kept his mouth shut and waited a while longer, there would have been no opportunity for Joffrey to seize the moment and give a public order for execution that the saner heads on his council could not block. Thus, after Robb's victories in the Whispering Wood and the camps besieging Riverrun, Ned would have been untouchable. With Jaime as a hostage, Robb would have had a vastly superior negotiating position. After all, a father who stands between Robb & lordship of the North might not be the best hostage, since he could bluff the Lannisters and scoff at their threat and demand greater concessions to let Jaime go. After all, if someone had Robert hostage, and demanded that Cersei & Joffrey release their most dangerous enemy in exchange for the release of Robert, how inclined would that mother and son be to give up their hostage and their power to resume their secondary places in his entourage? Even if Robb & Catelyn would be bluffing when they told Cersei to go ahead and kill Ned, the Lannisters would have to take it seriously (also Tywin had been similarly intractable regarding his own father-in-law back when he was merely the heir to Casterly Rock).

In any event, the Starks could have obtained Ned's freedom, had he been a little more patient, and with Ned alive, a coalition of Lannister enemies becomes more likely. At the very least, with his guidance, the Starks, and by extension the Tullys, would certainly have aligned with Stannis, and given Renly & the Tyrells pause, or at least make them willing to cooperate to destroy the Lannisters before quarreling over the throne (part of Renly's willingness to blow off Robb & Stannis was due to their individually weak positions, and Renly's belief that he could crush either one with ease at his leisure). An alliance between Ned & Stannis would have made the alliance against the ironborn and Others that Stannis is currently trying to build much more likely, and even stronger. They would also be more amenable to returnind Daenerys to the throne, as her legal claim would have meant more to those two than any other of the Great Lords. All in all, a world of hurt descended on Westeros, because Ned could not bear to hear a threat to his spoiled treacherous daughter.