Make sure you have no Java, JVE, JS2E listed in your Add/Remove programs in the Control Panel. If any Java is listed highlight and then Remove. Once removed make sure to reboot your computer.
If none were found still reboot the computer. Try running the install again.
If still problems visit the below link and download the manual offline version for Windows XP. This fille will be a lot larger but shouldn't require Internet access.
Try running the file. Again keep the name of the file the same name, so you'll be double-clicking the jre-6u5-windows-i586-p-s.exe file on your desktop. Don't rename it.
Finally, if that still doesn't work since this is an offline version, reboot the computer and boot up in Safe mode. Additional information about booting into safe mode can be found on the below document.
http://www.computerhope.com/issues/chsafe.htmIn Safe mode try running that same above file.