Mirroring is a specific concept in SQL Server but I am guessing you meant it generically because you cannot mirror to anything but another instance of SQL Server and you can only mirror an entire database, not just single tables.
In terms of SQL Server you are probably after 'Replication' which can be setup at the table-level. Sadly however, replicating data from SQL Server to MySQL is not something Microsoft officially supports. You could look for guidance online to see if someone else has achieved it, or look for third party software to help.
Rather than trying to replicate data to MySQL, can you simply connect to SQL Server from Joomla? SQL Server is a far more robust platform than MySQL. Think of it as a LASP stack ;-)
There are no special teachers of virtue, because virtue is taught by the whole community. --Plato