But I can only download the page and see the login form using the web browser (Internet Explorer) - which, of course, makes no use of the perl modules that are installed.

Using perl, I get no further than you.
Anyway, the versions of the various perl modules I'm using are:

C:\_32\pscrpt>pmver WWW::Mechanize
1.72

C:\_32\pscrpt>pmver Crypt::SSLeay
0.58

C:\_32\pscrpt>pmver Net::SSLeay
1.48

C:\_32\pscrpt>pmver LWP::Protocol::https
6.03

C:\_32\pscrpt>pmver LWP::Protocol
6.00

C:\_32\pscrpt>
[download]

Cheers,
Rob