May I suggest another first-hand perspective?
Livni should not have said your version of the sentence, because it is not true - the current government that came to power shortly the aforementioned negotiations was markedly more right-wing than the one in which Livni served.
Look, I don't like the current government. I didn't like the Olmert and Livni government either. But to claim that it was not democratically elected, or that it does not reflect the will of the people, would be misleading. A small majority of the Knesset seats is held by parties clearly identified as the "right wing" - Likud, Yisrael Beytenu, Shas, etc. Many, and possibly most of the voters of these parties outright object to the creation of a Palestinian state in the west bank. The rest of these voters want a government that is "tough", and do not care much if the Palestinian state is created or not - they're not sufficiently interested in it to vote to any party that is seriously committed to a 2-state solution.
It's true that Yisrael Beytenu ostensibly claims to support some form of a 2-state solution - as does parts of the Likud - but you can see from the lack of willingness to negotiate just how uncommitted they really are. From decades of experience of living in this country, I can assure you that almost no right-wing voter is going to protest the lack of peace talks.
Now, if you claimed that the elections do not represent the will of the people because the west-bank Palestinians don't get to vote, then you'd be somewhat correct (some of the Palestinians do have their own government now, of a sort). But as long as you're talking about Israeli citizens, the current government has very firm implicit support for its line of no-serious-negotiations. The main pressure they feel from the electorate is not to cede any territory.
A final note - none of them is a psycopath. It's just that these governments and their voters do not care much about democracy. The way some of them see it, the Arabs were pretty much ready to murder every Jew in this land back in 1948, and to repay those Arabs with anything less than genocide is being "generous". I don't agree with that point of view, but I can see where they're coming from.