The error here, as Loinen indicated, is the idea that the value of a product "is determined by the amount of human work consumed to produce them." First, I'd like to say something about subjectivist economics, which I've only just begun to study but which I find provides fascinating solutions to problems like this. Secondly, I'd like to address the idea that the worker doesn't get any of the profits, which is, while remotely possible in theory, generally false in practice, at least in developed countries.
The subjectivist school of economics, of which Ludwig von Mises was the foremost proponent*, argued that, in fact, nothing has any absolute (market) value. No commodity, service, or action. Rather, value is negotiated between parties in order to yield a satisfactory arrangement.
In the case of the sewing machine, it is not clear that the profit derives its value solely from the labour. After all, it is not the machine which produces the surplus. It is indirectly the work of the capitalist to employ the machine, which is effectively potential profit – ie., speculation. In fact, the labour would be as unable to produce the profit without the proper equipment, as the equipment alone would be without labour to operate it. This is because we have already assumed, as a premise, that labour without the equipment produces two units, not twelve.
Therefore, the profit can be split. How do we determine how it is split? This is not something that we can determine with formulas, because it is an affair between the service-provider (the labourer) and the service-consumer (the capitalist) and a whole load of factors, including the personalities of both parties, will come into play. However, there is one basic economic consideration: the discretion of the controller of the equipment. This service is more scarce than the labour.
The reality of owning capital is that it is not a magical cornucopia that showers wealth on the owner regardless of how it is employed. Someone has to decide what is going to be produced in the first place, and they must bear responsibility (and liability) for that decision. This person is the entrepreneur. If the entrepreneur is the one to take first action and employ a certain kind of capital for a certain line of production, it is by that person's speculation that a product exists to be sold – because although an entrepreneur without the help of labourers has only ideas and no product, workers without someone to anticipate the circumstances of consumers and bear responsibility for that speculation are relying on their own abilities to anticipate what people want. And the ability to successfully anticipate what huge masses of people want is rather rare.
Does this mean the capitalist "should" be paid more? This is a matter for philosophy, not economics. Suffice it to say, the ingenuity of the capitalist gives him a negotiating power that the labourer can't match. Therefore, we can naturally expect that the majority of profits will go to him, because he will associate only with those labourers (from among millions) who will agree to permit him to take the lion's share of the profit.
Some profit will still go to the labourer, in the form of wages. This is profit, because it is more valuable than that which the labourer could produce on his own (ie, doing subsistance farming and hunting with his bare hands in the wild).** His share of the profit will depend on the capitalist's inability to replace him. The worker has only one significant tool to negotiate value here: the strike. Assuming that the worker is not legislated to work, and assuming that the capitalist is not legislated not to replace him, then some genuine value negotiation can take place.
Marx, of course, was not a subjectivist. He sided with the proletarian through and through, and was possessed of the medieval peasant idea, transformed into a Victorian ideal, that work had innate value. There is no way this could ever be established to be so. No product has innate value; market value must be negotiated among participants in a market economy. This includes labourers as well as capitalists.
* It should be noted that, although Mises is often used by libertarians as patron saint of laissez-faire capitalism, his magnum opus, Human Action, does not rely on libertarian casuistry to make its arguments. How could a self-professed "subjectivist" justify reliance on casuistry to make an economic point?
** If he's worse off than he would be farming (as is often the case in modern third world countries, where the governments have taken away the farmland of the people), then the situation is a little different. Also, the wealth you can accumulate hunting and gathering is as much a function of the environment itself as it is of labour; in an area rich with grains, fruit plants, and small game, you can have a relatively affluent hunter-gatherer society. Not so in the Arctic or the desert. Unfortunately, in almost all temperate and subtropical areas man depleted most of the game before history even began, and multiplied well beyond the point where he could sustain himself for even a single generation on hunting and gathering alone.