The project that houses the migrants should be mentioned. Make certain you own an
the project's file that has the proper connection string in it.
The connection string used when executing an application that has been divided up over numerous projects is the one from the project that was first launched. The connection string used during migration is that of the project containing the migrations.
The connection string had to be added twice when I performed a configuration identical to that. While a little odd, it works.
By providing it as a parameter, you may avoid handling it in app.config:
Update-Database -Verbose -ConnectionString "CONNECTIONSTRING" -ConnectionProviderName "System.Data.SqlClient" -StartupProjectName WEBSITE_PROJECT -ProjectName MIGRATION_PROJECT
Easy-peasy if you like typing for hours on end.