Cannot Connect to SQL Server 2008 R2
Luckly I also have the same set up on my desktop. I have installed first default instance and then Sql Express. Everything is fine for me for several days. Then I tried connecting the way you trying, i.e with MachineName\MsSqlServer to default instance and I got exctaly the same error.
So the solution is when you trying to connect to default instance you don't need to provide instance name.(well this is something puzzled me, why it is failing when we are giving instance name when it is a default instance? Is it some bug, don't know)
Just try with - PC-NAME and everything will be fine. PC-NAME is the MSSQLServer instance.
Edit : Well after reading your question again I realized that you are not aware of the fact that MSSQLSERVER is the default instance of Sql Server. And for connecting to default instance (MSSQLSERVER) you don't need to provide the instance name in connection string. The "MachineName" is itself means "MachineName\MSSQLSERVER".