The 'mirror' function of LWP::Simple only acts on a given URL. It does not "spider" the site, following links and image references. If that's the behavior you want, you will need to use something like HTML::Parser to parse the results of your 'mirror' (assuming it returned a value indicating something changed), pull out the links and image references, and then 'mirror' each of those.

Surely there is a pre-written script or module that will do this though...? I don't have any experience with any, but I imagine spidering a site for the purposes of mirroring is something that's been solved several times over already. If you're only worried about one page, I'm sure most of these have a 'depth' option that you can simply set to 1.