The velocity of money can vary, aggregate demand matters, and the multiplier is real. Let's get those preliminaries out of the way. That all said, this is one of the least accurate chapters in Keynes's General Theory. To pull out one key quotation (pp.116-117, in section II):
It follows, therefore, that, if the consumption psychology of the community is such that they will choose to consume e.g., nine-tenths of an increment of income, then the multiplier is 10; and the total employment caused by (e.g.) increased public works will be ten times the primary employment provided by the public works themselves…
Empirically a typical estimate of a multiplier might be 1.3 or 1.4, not 10, not even in a deep slump. (Valerie Ramey points out that the key issue in estimating a multiplier is to determine when the fiscal innovation actually occurred; this is not easy.) One theoretical problem in generating a high multiplier is this. Say you have a debt-financed increase in government spending. You can get some dollars out of low-velocity pools into high-velocity pools on the first round of redistributing the spending flow. Do not expect complete crowding out and so nominal aggregate demand can increase, thus boosting output and employment. But the second and third round effects of the redistributed money are usually a wash and the boost to velocity dwindles. Why should it stay in a high-velocity sector of the economy?
It is common in the GT that Keynes confuses marginal and average effects and, for all of his explicit talk about average and marginal in this chapter, he is making one version of that error again. Rothbard and Hazlitt are not in general reliable critics of Keynes, but
they do have a good reductio (see chapter XI) on crude interpretations of the multiplier and that is what Keynes is serving up here. You can't just take a partial derivative of an accounting identity and call the result a causal relationship.
In addition to velocity/spending effects, there are also multiplier effects through real production. The most insightful analysis of supply-side multipliers comes from the work of W.H. Hutt.
The multiplier is a legitimate concept but often it is overestimated in its import. This chapter in Keynes is a step backwards from Richard Kahn, the father of the multiplier concept.
Don't forget Alex's comments on fiscal policy and velocity.