did you see that Apple rereleased the
Java update ? You should now be able to run Software Update and install the new java and QT OK on 10.2.8 (works here now).
Before you downloaded a fresh installer and reinstalled LW, did you find and delete the old LW prefs too? Might be worth a shot--you can find them by searching for 'limew' with the Finder.
If you still can't launch, post back. There are alternates to the usual installer, but not tested as well as 3.6.15.
Cheers