It looks like the Java installation worked and LanguageTool should work, too.
Please check the LibreOffice version number (Help->About). If it starts with a 4.x, you have an older version that might not work with LanguageTool.
Shameless plug: now that you have Java installed, you also might give my
LanguageTool Sigil plugin a try. Download the standalone
LanguageTool version, extract it to an easy-to-remember folder and follow the installation instructions in the
Plugin thread.
(In order to install the plugin, you'll need to know where
languagetool-commandline.jar is located as you'll need to select it when the plugin is run for the first time.)