I am using TeXstudio with Language Tool under Windows 7 without any problems. The entry in the wiki is not valid for the newest version. You need to use the languagetool.jar. Then everything should work. All the other files you mentionend, existed in previous version of the tool, but are now removed. See the changelog.
Unzip the downloaded package and open languagetool.jar. Open "Text Checking"-> "Options", tick "Run as server on port" and set it as 8081.
Open Texstudio and click on "Options"->"Configure Texstudio".
Check the "Inline Checking" in "Editor" section.
Make sure that "Show advanced options" in the lower left corner is checked, and modify the "LanguageTool" in "Grammar" section as below.
Specifically, set "Server URL" as "http://localhost:8081", "LT Path" and "Java" respectively as the paths of "languagetool-server.jar" (Old Version "languagetool.jar") and "java.exe".
To test this, (re)start texstudio, and check whether the Languagetool is running on the desired port by navigating to http://localhost:8081/ on your machine.
This is a recent and more detailed solution for Windows users.
Make sure the last version of TeXstudio is downloaded from its website.
Download the English dictionary (*.oxt) from here to the directory C:\Program Files (x86)\TeXstudio\dictionaries.
Download languagetools (*.zip) from its website (I used D:\Applications\LanguageTool-4.8 as its directory).
Navigate to LanguageTool directory, then open languagetool.jar. Go to Text Checking > Options, then check Run as server op port and type the port number 8081. Click OK and close.
Open TeXstudio and go to Options > Configure TeXstudio > Language Checking while making sure the Show Advanced Options is checked.
Click on Import Dictionary... and navigate to C:\Program Files (x86)\TeXstudio\dictionaries, then choose the dictionary *.oxt you downloaded.
Make sure the default languae is selected (e.g. en_US), and the thesaurus is selected from the same directory (e.g. th_en_US_v2.dat).
Check Start LanguageTool if not running.
Set the values Server URL, Java, LT Path and LT Arguments to http://localhost:8081/, java (as recommended here), <directory of LanguageTool*-*>\languagetool.jar and org.languagetool.server.HTTPServer -p 8081 --languagemodel --allow-origin "*" (as recommended here), respectively.
In order to make sure everything is fine, click on Help > Check LanguageTool in TeXstudio. The output should be something like this