Kaf is connected to the Hebrew word Keter, crown, which in Kabbalah is connected to those aspects of the soul that are above intellect: Emunah (faith), Ta'anug (pleasure), and Ratzon (desire). Kaf also means the palm of the hand, or a spoon, both of which are slightly bent in order to serve as a receptacle, a kli (which is also with the letter Kaf).
Lamed is the root of the words Lilmod (to study) and Lelamed (to teach). The Lamed is particularly connected with the Oral Torah, the part of the Torah which was never intended to be written down, but instead was transmitted orally from teacher to student.
In Kabbalah, the Day is connected to intellect, while the Night represents that which is above intellect, particularly Emunah, faith. These concepts can be found in the verses each of the two sing in Perek Shirah:
The Day is saying: "Day to day utters speech, and night to night relates knowledge." (Psalms 19:3)
The Night is saying: "To speak of His kindness in the morning, and of His faithfulness by nights." (Psalm 92:3)
The Hebrew word for knowledge is Da'at, which in Kabbalah is interchangeable with Keter. Faithfulness above, in Hebrew, is Emunah.
Teveth is a time of the year where the nights are particularly long and cold, yet around this time is also when the nights slowly start getting shorter again, and the days start getting longer. Our sages tell us that long nights were given in order to study Torah. Traditionally, the Oral Torah is what is studied by night, while the Written Torah is studied only during the day.
Shevat is particularly connected to all of the above. As explained in Book 1, Shevat is connected to the Tribe of Asher, which stands for Ta'anug (pleasure). Shevat is also very much connected to Emunah, given that we celebrate the New Year of the Trees, Tu B'Shvat, in the middle of the winter. Finally, Shevat is connected to the Oral Torah. Moshe began teaching the Oral Torah to the Jewish People during this month, and the Chidushei HaRim teaches that all insights in the Oral Torah for the entire come to a person during the month of Shevat.