"The exile of the Shekhinah" is not the Shekhinah's absence but its involvement in other, unsanctified matters.
The sages state that "when [the Jews] were exiled to Babylonia, the Shekhinah went with them.
The Shekhinah is with them, even in exile, but then it is concealed,clothed in sackcloth–in nonholy matters.
When it wears this garb, it can turn against even holiness itself.
A Hasidic rebbe once said that there are three stages of the exile of the Shekhinah.
The first is the exile of the Jews among the nations.
The second, and harsher, is the exile of the Jews among themselves.
And the third, and most grievous of all, is the exile of an individual Jew within himself.
–Rabbi Adin Steinsaltz
From Learning from the Tanya, p. 131, by Rabbi Adin Steinsaltz