In my experience, some classes just suck. 
I'm at a low level tech high school as well, and it doesnt matter what I do in a few of my classes. At least half of them have no interest in what I'm doing. (or that interest quickly wanes when the work part of the lesson comes around) They like me and try to talk to me outside of class, but in the classroom theyre practically comatose.

I've tried everything.  Various interesting lessons tailored around things that they're into, games, punishments, rewards, group work, etc. 
I can be, and have been the hard ass that would force them to do stuff...but, that got tiring real quick. I don't like having to come to class everyday and put on the angry face. It's a waste of my energy, and just sours the rest of my day.

If I just have a couple of bad/zombie students in class, then I'll deal with them so that they don't bring down anyone else...but when it's half, or a majority. then my options are more limited.

I've done things like reward the students that participated. (that usually gets the sleepers attention and theyre awake for the next class hoping to get something)
And if we're doing some writing assignment, i'll let the ones that actually did it leave early...those that didn't get to stay till their done---(I only do that with my class before lunch and the last period class to avoid issues with other teachers though)
Things like this only seem to work for a week or 2 though.

They do like team bomb games...that usually has the whole class up and into the lesson, but I don't want to overuse those for fear that they'd get tired of them.

I'm at a technical school too. My students are well behaved in the main, as they must be to avoid being beaten!

I also have issues with maintaining more than half of my second grade classes' attention. The best results I've had is with problem solving tasks such as solving riddles and sentence jumbles. Pronunciation and idiom lessons work too. Videos also work to grab their attention for short periods. Heavy writing classes are impossible. Some classes enjoy role play activities where they can muck around in front of the class.

I've also resorted to being the tough guy at times and in general it has worked. It hasn't increased my popularity any but it works (most students still like me however). Since the principal and VP cruise the corridors, I cannot risk sleepers in my class.

 

I've never taught HS, & I'm not 100% certain what constitutes a "technical" school in Korea, but I do have an idea.

If you have access to it, or have any sort of budget for teaching materials, try to get your hands on some Lego Technic, Mechano sets, or similar.

You could start by showing some videos of amazing things that others have built. I'm pretty sure you'll get their attention if you show a video of a working gatling gun made out of lego, a robot, or any of the many other cool things people have made.

The first classes could be just about getting them engaged by playing around with the building sets, but later, if at all possible, you could try having them follow a set of instructions with some pre-taught English words mixed in (you'd obviously need help from a KT to do this), or have them build something simple, and then describe how to build it to a friend, (without one another seeing) and then have them compare the two to see if they match.

Of course, it could easily turn into a situation where they are all just messing around with the building sets, but at least they're not sleeping. Maybe use the building time as a reward after a brief English lesson at the beginning. Good luck.