Vivekananda considered it to be the Kali Yuga. But Vivekananda also points out that 'yugas' are only spoken of in the Smritis, they are not spoken of in any of the Srutis. So, what 'yuga' you consider it to be, whether or not there are yugas, what the exact date is, is not important.
It is important to understand what Vivekananda is talking about in the quote you gave. In it, Vivekananda says that "special stress" should be made on Karma - he does not say that it is THE way for the Kali yuga.
Vivekananda says (Complete Works V3, pp 221-2; and here under the heading Lectures from Colombo to Almora, sub-heading My Plan of Campaign) http://cwsv.belurmath.org/volume_3/vol_3_frame.htm
So every improvement in India requires first of all an upheaval in religion. Before flooding India with socialistic or political ideas, first deluge the land with spiritual ideas. The first work that demands our attention is that the most wonderful truths confined in our Upanishads, in our scriptures, in our Purânas must be brought out from the books, brought out from the monasteries, brought out from the forests, brought out from the possession of selected bodies of people, and scattered broadcast all over the land, so that these truths may run like fire all over the country from north to south and east to west, from the Himalayas to Comorin, from Sindh to the Brahmaputra. Everyone must know of them, because it is said, "This has first to be heard, then thought upon, and then meditated upon." Let the people hear first, and whoever helps in making the people hear about the great truths in their own scriptures cannot make for himself a better Karma today. Says our Vyasa, "In the Kali Yuga there is one Karma left. Sacrifices and tremendous Tapasyâs are of no avail now. Of Karma one remains, and that is the Karma of giving." And of these gifts, the gift of spirituality and spiritual knowledge is the highest; the next gift is the gift of secular knowledge; the next is the gift of life; and the fourth is the gift of food. Look at this wonderfully charitable race; look at the amount of gifts that are made in this poor, poor country; look at the hospitality where a man can travel from the north to the south, having the best in the land, being treated always by everyone as if he were a friend, and where no beggar starves so long as there is a piece of bread anywhere!
In this land of charity, let us take up the energy of the first charity, the diffusion of spiritual knowledge. And that diffusion should not be confined within the bounds of India; it must go out all over the world. This has been the custom. Those that tell you that Indian thought never went outside of India, those that tell you that I am the first Sannyasin who went to foreign lands to preach, do not know the history of their own race. Again and again this phenomenon has happened. Whenever the world has required it, this perennial flood of spirituality has overflowed and deluged the world. Gifts of political knowledge can be made with the blast of trumpets and the march of cohorts. Gifts of secular knowledge and social knowledge can be made with fire and sword. But spiritual knowledge can only be given in silence like the dew that falls unseen and unheard, yet bringing into bloom masses of roses. This has been the gift of India to the world again and again.
So what he means by special stress is that Karma, in the form of giving should be done. What he is saying is that no matter what you practice--whether its bhakti, raja, jnana, or more ritualistic karma, or a combination of all or some of them--that you should also be practicing the karma of giving, especially the karma of diffusion of spiritual knowledge. Lord Krishna says (Gita, XVIII. 68-69):
He who, with supreme devotion to Me, teaches this deeply profound philosophy to those who are devoted to Me shall without question come to Me.
There is none among men who can do anything more pleasing to Me than he, nor shall there be another on earth dearer to Me than he.
Follow Bhakti (Vivekananda's guru, Ramakrishna Paramahamsa said that Bhakti was the easiest path in the Kali Yuga), but also pay heed to giving.