Props to vaughan_stanger
for convincing me to give ljArchive another try.
Whereas yesterday it failed bizarrely every time I ran it, today it is working. You need at least one good sync for the tool to start working properly, so you may have to try many, many times...
Here's the link:LJArchive
It's not perfect. I don't think it downloads images locally (seems to leave them on LJ, even when I export to an HTML file), but it is a well-organizaed backup of all the text.