Okay, I installed the drivers, restarted twice. Didn't fix anything, but I did indeed have factory-default drivers that sucked, so nothing of value was lost.
So then I uninstalled Leadweks again, then I used RegEdit and CCleaner to muck around with the registry and some files, cleaned my computer of all files and keys related to Leadwerks.
And, for good measure, I restarted and reset all my NVidia settings to default.
Whatever I did after installing the drivers seems to have worked, as now the Editor starts up as normal.
Thanks for the advice, you peoples.
And now on to my second problem: why is the bug tracker locked for non-paying members?