Flesh is a Java application designed to analyze a document (plain text, rich text, Word documents, and PDFs) and display the difficulty associated with comprehending using the Flesch-Kincaid Grade Level and the Flesch Reading Ease Score.
I ran a test on a Word doc, and the results differed considerably from MS Word's own Readability Analysis. This tool is only of use if it is accurate; and as a professional Technical Author I'm inclined to trust the millions invested by MS. I hoped to find something that a client could use on Linux, but I'm not sure that I can. It doesn't appear to accept .docx files. There is no support offered. Suggest you try it and see, take pot luck.
I use extensively the command line version. It's very useful to improve text readability.