Could not start the SQL Server Agent (SQLEXPRESS)
SQL Server Agent is not available with Express. It does install but that is just the false hope Microsoft puts in for some reason.

If you are seeing the "(EXPRESS)" within SQL Server Configuration Manager it was likely installed at some point in time and an attempt to remove it left some things behind. Express edition can get installed if you happen to have installed Visual Studio 2010 for instance.

Since you see the "(MSSQSERVER)" it would tell me that the default instance is the full version (Standard or higher) of SQL Server. There are a few ways to "break" into the default instance you have on your machine/server. However it depends on your environment as to what method you want to use.