What day is discussed?
Radak: It is the day that you will be exiled from your position. It is the time (not literally the day), like "ba'Yom ha'Hu Shoresh Yishai Asher Omed Elyakim l'Nes" (11:6).
What is the meaning of "v'Karasi"?
Radak, Malbim: I will put in Chizkiyah's heart to put Elyakim in place of you.
Targum Yonasan: I will raise. Radak - i.e. I will aggrandize him and appoint him over Beis David. Also Targum Onkelos of "Re'eh Karasi v'Shem" (Shemos 31:2) is like this. Also "Shalishim u'Keru'im" (Yechezkel 23:23) are great people; "Keru'ei ha'Edah" (Bamidbar 1:16) are great people of the congregation who are called for all counsel.
Who is Elyakim?
Rashi: He was [appointed] over the Bayis when Sancheriv fell. It says that he was [appointed] over the Bayis when he went out to Ravshake (36:3).
Radak: We find in Sefer Melachim and in this Sefer "Elyakim Al ha'Bayis" and "Shevna ha'Sofer" (Melachim II, 18:18). This was when Sancheriv fell. Or perhaps it was earlier; he rebelled and conspired with Melech Ashur. He concealed this, but Hashem revealed it via the Navi - "Ba El ha'Sochen ha'Zeh" (15). Elyakim was appointed Al ha'Bayis, and Shevna remained Sofer. Even so, Shevna tore his clothes when he heard Ravshake's words; he covered himself, and pretended that he was pained.
Radak (22): Perhaps he was Kohen [Gadol]. We find (Divrei ha'Yamim 1:5:39-40) "v'Shalum Holid Es Chilkiyah v'Chilkiyah Holid Es, Azaryah". Perhaps Azaryah is Elyakim; he has two names. We find that Azaryah was appointed over Beis Elokim in the days of Chizkiyah.