There might be out-of-the-box tools for this, especially for wordpress, but my approach would involve writing a teeny bit of code and then either host the tool somewhere or just run it on your computer (since you're the only one interested in the result) to save 100% of the cost.
If I were to build something like this for myself (and I might, sounds like a great idea), I'd use a quirky little Python package called scrapy ("The world's most used open source data extraction framework") and simply go through all the "<a>" tags on your site (I see that pretty much all the information is on a single page which makes things much easier), call the "href" or link/url on it, and see what comes back, creating a list of broken links and logging them for the user to review afterwards.