What's the mistake?:

"Annyeonghaseyeo! Lee Kiyeon imnida! Please take care of me!!" I bowed and rushed to my chair. 

"Kiyeon?"

"Ne?!" I looked to the teacher.

" Can I see you after class?"

"Ah, wae?"

"Don't ask."

"Ne..."

Here is an example of Konglish. Which is Korean + English. I am guilty of using konglish and I'm sure many fanfics include konglish. This isn't much of a criticism but more of an opinion section. 

In my god honest opinion I don't mind konglish as long as its thinks like, "wae", "annyeong", "imnida", "ne", or other most commonly known korean words.

Although there are a lot of people out there who will absolutely go livid if they even see the words listed above in your story. When I first started writing my stories here on AFF I used ne and annyeong and wae a lot. As I began to mature and get to use to things here on AFF I got out of the habit of saying wae and annyeong and other korean words in my stories to make my stories seem more mature. Plus, even though those are basic korean words, not many people may even know what they mean.

There is a right and wrong way to use konglish though if you so decide to use it. 

The right way would be to use simple korean that most kpop lovers will know. such as the words above but not pertaining to. (other words like; sarang, saranghae, dorawa, kaja, ani, noona, oppa, unni, hyung, palli, umma, appa, omo, aigo, aish, mwo)

The wrong way to use konglish, or korean within a story is by using long/big words/sentences that no one may have a clue to even though you may know it, it doesn't mean any one else may know it:

"Omo! Neuj-eoss eo!" which means "Oh no! I'm late!"

If you have to translate your sentence because people won't know then you shouldn't even use that korean phrase.

However do know that not every reader who reads your story will like the konglish you mix into your story.

So how do I fix these mistakes?:

What do you suggest?; In my opinion I suggest not doing it at all. Mainly because it's not gonna harm you not to use korean in your story but it can harm you by having annoyed readers or people who don't subscribe because they are picky like that.

Korean names are okay?; Absolutely.

What if my character is learning korean or they are teaching someone korean?; This is different. If your character is learning or teaching korean then it's part of your story however "writing" the korean words as a response like "annyeong" or "wae" is not really a part of your story you are only making it apart of your story.

So what about oppa, dongsaeng, noona, hyung, and unnie?; These words I consider to be relatively okay no matter what.Even though I don't use korean responses any more I still use oppa and noona etc because a lot of people consider them to be cute! Its really up to you what your preference is and what you want to do to please your readers. Or if you don't care about really who reads or what your readers think then do as you please!

Extra tips:

If you have to type out your story and you have to stop for a sentence because it's in korean just to translate it to your readers then take out that korean sentence and write it in english. Interrupting your story (with brackets to explain something) can be distracting, tacky, and annoying.