I have Microsoft SQL Server 2008 installed and running but I can't seem to see
COMPUTERNAME/MSSQLSERVICE at all.
COMPUTERNAME/SQLEXPRESS but that's not the right one.
So looks like you need to check all instances of sql server running and available. You can check then what if the name is correct or the service is running.
Two ways you could approach this:
If you have Sql Server Management Studio installed:
The chances are that the full SQL Server instance on the machine is the default instance, therefore to access it you don't need to specify an instance name.
If unsure, use OSQL to list instances on the machine.