Given a set of Java source code files, how can I compile them into one or more JavaScript files that can be used with hand-crafted JavaScript?
GWT is one option, but
Given a set of Java source code files, how can I compile them into one or more JavaScript files that can be used with hand-crafted JavaScript?
There is no direct correlation between both the built-in Java API and Java language features, and those of JavaScript. So any attempt at creating a "converter" is going to be incomplete. As a fundamental example, Java classes don't have a direct corresponding JavaScript idiom.
Whether or not an incomplete conversion tool will work for your use case is impossible to know without the source code.
That said, my suggestion to solving your problem would be to first attempt to use GWT: set up a demo project, drop in the source of your library and call it from the JavaScript side and see what GWT outputs in it's .js file. Some of the other tools suggested by other posters here are definitely worth checking out as well.
If that is fruitful and gets you part of the way, great.
From there, you'll need/want to do the remainder of the conversion by hand. After all, assuming you want the code to actually function correctly, a manual review would definitely be in order. Some unit tests being converted along with it would be ideal as well.
You don't state how large the source of your project is, but if it's small (let's say less than a thousand lines of code), even a complete conversion by hand shouldn't be extremely difficult. If it's much larger than that, I would suggest reviewing if you actually want that as JavaScript code anyway.