I've never worked with Net::SSH::Perl, but looking through its code, I'm getting a feeling you may need to call $login->login() before authenticating. This seems counterintuitive to me, but apparently login() (well, _login()) is what sets the AuthMgr on a Net::SSH::Perl::SSH2 object in the first place.

I'm afraid I can't try it, but it should be a starting point for further investigation.