I think this video is good —
— but somewhat misleading. It makes it sound like Obama's stimulus would not have failed if the money actually reached the intended (or, rather, advertised) targets and were not wasted or did not create unintended consequences. (I differentiate between 'intended' and 'advertised' because I believe, rather cynically, that the Federal government all along knew that the money would go to its buddies, in the usual crony-capitalist fashion.)
But the deeper point is that even if that were to happen, this money would still fail to 'stimulate' the economy. What was wrong with the economy is not that there was not enough money or spending or that it just needed a kick in the right piston to get the machine working again (see here for the rebuttal of this fallacy). The problem was not with lack of investment — but, rather, with malinvestment: during the boom phase of the business cycle, money was invested into the wrong industries (see here why this happened).
During the bust phase, the economy 'discovered' that there was after all no demand for the long-term, 'capital goods' businesses (such as construction) into which the investments had previously flowed (encouraged by the artificially low interest rates). This discovery led to contraction of the suddenly failing businesses (or, rather, businesses whose failure suddenly came in the open): their stocks dropped, they had to fire many of their employees, and even declare (or come close to declaring) bankruptcies.
This process of contraction is not only natural, following the malinvestment of the boom phase, but is in fact crucial for any 'recovery'. In order to get the economy back on the road, it was — and still is — necessary to allow the businesses that became too engorged with the malinvested capital to shed it and let the capital flow to the businesses that actually have the support of their customers (because they are successful at predicting what their customers need and want to spend money on right now). This could happen through the successful businesses buying up now-cheap equipment and land or office space of the failing businesses and hiring the suddenly unemployed workers.
By keeping the failing businesses alive or by taxing the successful businesses and redirecting the tax money towards roads or public schools, one does not reverse the problem of malinvestment. And by making interest rates even lower, thus encouraging further long-term investment unsupported by the public's spending time preference, the FED is making the problem worse!