Here are a few thoughts:
It isn't clear who is talking in the first two quotes. I assume the first one is your teacher and you are the second? You might want to clarify or even omit the first quote. Are these internal thoughts or are they spoken aloud? Usually, if something is not spoken aloud, it would go in italics instead of quotes.
Italicize the Spanish words instead of putting them in single quotation marks (including telanovelas
heavy packet of papers
This is a bit redundant. The reader can assume that the packet contains papers.
going off the roof
through the roof
I am not sure why you have the single quotes. The only time you would use single quotes would be if you had a quote inside of a quote.
My adventure to conquer this new language began as I attacked Spanish with the incantations of memorization.
There's a lot going on in this sentence, and the meaning gets lost. I'd try to rewrite it and tighten it up a bit.
I could not depend on my mentor to guide me in this battle for her methods of teaching constantly confused me. Yet, my independence was a blessing in disguise as I learned my most effective ways to study.
This might come across as negative to the reader. It sounds like you are saying that your mentor couldn't teach and that you are better at devising study habits. Rewording this to indicate that you do not always learn by traditional methodology and overcome that obstacle with perseverance would put a more positive spin on it.
I would write the words in Korean to practice my pronunciation and repeated the definitions of words until I memorized them.
Your verbs aren't parallel here.
During my first listening comprehension tests, I was stumped by the meaning of a single word and the essence of the entire dialogue was lost to me. It was the same with speaking as I stumbled upon a word or a verb tense that I discontinued talking.
These two sentences aren't as clear as they could be.
Yet, these problems were part of the inconspicuous joy of learning a new language and every failure was a step closer to proficiency.
You'll need a comma before the word "and" because the second clause could stand as a sentence on its own.
There are a lot of strengths in this essay. I like the way you have used mostly active verbs, the variety in sentence structure, and the way you have incorporated interesting vocabulary.