The connection name 'LocalSqlServer' was not found in the applications configuration or the connection string is empty

Maybe there is a web.config somewhere that contains:

<connectionStrings>
    <clear/>
    ...
</connectionStrings>

You might also want to double-check that you're looking at the right machine.config. There are separate machine.configs for each Framework version, and also separate ones for the 32-bit and 64-bit framework.