You won't be able to run as administrator from control panel, which is exactly why LOTR says to download an offline java installer. The various java files are scattered all over your computer once installed, so if you want to run each and everyone of those as admin, then that method has not been tested.
You will find various java*.exe files in amongst other locations:
C:\Program Files (x86)\Java\jre6\bin
C:\Windows\SysWOW64 (is x64 version)
use search on your computer to find all locations. Maybe some are enough, maybe one is enough. We have no idea.
If the link LOTR provided doesn't work for you then try one here:
Download Java Runtime Environment 1.7.0.0 (32-bit) - FileHippo.com They should have most versions and the site is trusted.
Once you downloaded the relevant java, download it to Desktop, right-click the installer and choose run as administrator.
Of course after uninstalling the version you already have installed.