Mentioning unpublished research paper in Statement of Purpose/Resume
You can certainly mention "work in progress" or "submitted, not yet accepted" and the like in your materials. Just label things for what they are. Two years isn't really a long time for research in most fields, so I wouldn't worry much about the time. It is also natural, if you have been working in the field to immerse yourself in that for a while. 

But submitting the work is marginally better than not.