Publishing a work; not sure what should I do
First, you should ask someone famliliar with the field whether it is 


really unpublished
a good idea. 


Sometimes ideas are very old and might be published in books which are not available online, and therefor harder to find. Or you might have used the wrong terms for your search.

If after that you still think it is a unique and useful idea, you should try to publish it. The "useful" is important here, sometimes people are publishing new stuff which noone applied to the problem before just because it makes little sense with regards to the big picture of the discipline.

If your instructor did contribute to the paper and the research, then it should be on the paper unless he declares it is not necessary. I would ask my student to come with a paper outline first (so you could approach your instructor with a clear plan of the publication), we improve the sketch and then the studet writes the first draft which I as a supervisor try to bring it in a form which suits the standards of the respective conference or journal. Usually, this involves some feedback rounds and includes additional scientific discussion etc. . 

From your OP I got the impression your instructor might not be willing to put much effort in it, but sometimes people when they see someone is willing to go an extra mile. 

If you are on your own, try to read some books about scientific writing, there are lots of formal errors which can be easily avoided. 

I definitely encourage you to try to publish, it is a good experience and, if successful, is great for your CV.