Note that this only works on the core, and will spew a number of certificate warnings, making it less than suitable for programmatic recovery (in other words, my utilities will still be asking for the password on the command line for a while longer).

https://localhost/landesk/ManagementSuite/Core/ssl/information/DatabaseInformation.asmx?op=GetConnectionString

Be aware that you can't just paste that line into your browser, you have to browse to it.