Syllabification function for Turkish words
First of all you should know that the script doesn't syllabize properly for every word. For example if you give the word authenticated the function returns ['aut', 'hen', 'ti', 'ca', 'ted'] which is incorrect. The correct case would be ['au','then','ti','cat','ed']. However I do not know how to fix that.

Edit: The point made above doesn't stand since this script isn't for the english language.

Second you have a lot of ifs where you check if a character is a vowel like this:

if word[let_ind] in self.vowels:
        last_was_vowel = True
    else:
        last_was_vowel = False


They could be written better like this:

last_was_vowel = word[let_ind] in self.vowels


Or even better I would have written a function that checks if a letter is a vowel.

def is_vowel(self, char):
    return char in self.vowels


The first if in your for loop 

if let_ind != len(word) - 1:
    if word[let_ind + 1] in self.vowels:
        next_is_vowel = True
    else:
        next_is_vowel = False
else:
    syllab = word[keep_index:]
    syllabs.append(syllab)
    break


checks if the next one is a vowel except the last char.
It would be more good looking like this:

for pos, char in enumerate(word[:-1]):
    next_is_vowel = self.is_vowel(word[pos + 1])
    ...

syllab = word[keep_index:]
syllabs.append(syllab)


We enumerate through the word to get the position of the letter and 
the letter itself. The word[:-1] means we are going to iterate the whole word except its last letter.

Lastly you got 2 ifs here that do the same thing. You also don't need the keep_index.

if next_is_vowel and not last_was_vowel and syllab:
    syllabs.append(syllab)
    syllab = ""

elif next_is_vowel and word[let_ind] not in self.vowels and syllab:
    syllabs.append(syllab)
    syllab = ""


The could be easily written as one if like this: 

if next_is_vowel and syllab and not (last_was_vowel and self.is_vowel(char)):
    syllabs.append(syllab)
    syllab = ""


The place where the keep index was used can be replaced like this
from syllab = word[keep_index:]  to syllab += word[-1]

With these changes in mind the code looks now like this.

def is_vowel(self, char):
    """

    :param char: the character to check
    :return: bool depending if the char is a vowel
    """
    return char in self.vowels

def syllabicate(self, word):
    """
    :param word: The word to be syllabicated
    :return: The syllabicated list that contains syllabs
    """
    word = word.lower()
    syllabs = []
    syllab = ""
    last_was_vowel = False
    # we don't want the last character

    for pos, char in enumerate(word[:-1]):
        next_is_vowel = self.is_vowel(word[pos + 1])

        if next_is_vowel and syllab and not (last_was_vowel and self.is_vowel(char)):

            syllabs.append(syllab)
            syllab = ""

        syllab += char
        last_was_vowel = self.is_vowel(char)

    syllab += word[-1]
    syllabs.append(syllab)

    return syllabs