Have you checked the language settings in Libre Office?
This is probably a longshot, excuse me if I waste your time, I don't know if this will help but maybe the Language Tools needs you to choose the same language as the LanguageTool will use before you install it.
If you want to try this, start Libre Office Writer, Tools, Options, Expand Language Settings, Languages and change all settings to the Language your Language Tool will be used for:
Restart Writer and go back and check the settings, this is how it looks for me after restart:
Actually I now find I dont need to install the Language Tool because Spellcheck and Menus are now running in Swedish (my native languauge).
I don't know if Libre Office found the language files I downloaded and extracted automatically (I didn't install them, don't know how to).