Wordpress Importer "Failed to import Media"

I’m trying to rescue a WordPress site for someone who’s host dropped them.

I have a file-level copy of the WordPress files (for the theme) and the WXR export.

When I try to import the WXR file [Tools > Import], I get the normal author translation. Then it fails with a lot of Failed to import Media "..." errors. I was not expecting to be able to get the images (which have gone to the bitbucket in the sky). But I was expecting this to import the posts and pages, with just the text information. However, no pages or posts are imported, with images or otherwise.

I’ve looked at the XML file and this information is all there.

How do I import the pages and posts from this file (ignoring the images and attachments)