My coteacher told me I can pick students from 5th and 6th grade to be in my high-level 'extra' class.  Since I'm teaching the class alone I like the idea of choosing my students.  The problem is there are tons of students who excel at English.  6th grade just took an English test and the majority did great.  The test was probably too easy.

Last year the homeroom teachers picked the students...

Any ideas?

Yep, see if you can arrange for interviews during a free period or over a couple lunches.  These can double as a speaking test so you can get the levels you want, and can also allow you to see who would actually be interested in doing it. 

My coteachers always based the rosters off grammar and written tests and picked 20 to 30 students that way, but of course none of them really wanted to do it so only three to five ever showed up. 

There is no simple solution to your question. If it was a group of 4 students you could probably determine their level fairly quickly just by having a conversation with each one or observing them doing some task. The whole "what level is the student" is not a perfect science, although you could certainly find a list a linguist made that details what language skills the student should possess to be at a certain level and categorize them accordingly. I don't know many teachers that do this, usually there are simple cut offs like, is the student conversational, can they perform basic tasks and understand directions in the target language, etc. If we are talking about a classroom of 20-35+ students (the norm in Korean public schools) then there is just too much disparity in the students ability and you will have to get to know your students over-time and eventually you will be able to decide what they can handle and what they can't handle. I think the best thing to strive for is making groups that are of mixed ability and letting the high level students assist the low level students, but not do the work for them.   

I offered to have an unpaid/free English club along these lines and picked some students that were really really excited and I was shot down after I was told to ask them if they wanted to join. School's reasoning was that it was unfair to discriminate among the good students at high level and any other student. Jealous that you can actually have some sort of leveling at your school, AND it sounds like you are paid for this class too.

Personally if I was allowed to pick I wouldn't worry so much about skill level, but more on desire to be there. I really don't want any students who don't want to be in my class.

This, and only this, should be your deciding factor. You'll find in every school that good grades do not equate to good behaviour or motivation: sometimes it's just access to hagwons or whatever that determine a student's grades.

I find that most of my top level students in each class are also the biggest pricks in class. Unlike back home (Well, Australia in my case), good grades and intelligence are something to be proud of here, and it can often equate to arrogance and ego.

If I had it my way, I would never sort classes by grade (A, B, C, D or whatnot), but by motivation. How much English do they want to learn?

A harmonious class doesn't necessarily mean all students have the same level. A harmonious class is one in which the students are working together, not against each other. Some could be low, some could be high. Students that work together, and teach one another, are the strongest learners.

I remember some quote about people remembering 10% of what they hear, 50% of what they write, and 80% of what they teach or something to that effect.