Very good question but tricky answer.
Do you want to clean the existing compiled files OR also regenerate compiled files?
In the first case, you could simply delete any files located in your wp-content/uploads/wp-less directory. The compiled files will be regenerated on the next displayed page.
However it may not work if any server-side cache system is enabled and has not been flushed too.
At the moment, calling a page embedding a LESS file is the only way to recompile a LESS file. WP lacks of a CLI interface to process administrative tasks. An HTTP query could do the trick but it should be secured (secret URI known and (re)generated by an admin).
In the future, for an admin page feature, it would be necessary for the plugin to know about the enqueued LESS files at another moment than the runtime to display those informations. Prior to that moment, it will be a little bit hacky.
Otherwise, flush all caches, css compiled files and launch a browser to generate anything needed (to simulate visits and reduce the payload for the next real visitor).
