How do I disable TLS 1.0 without breaking my IIS/ASP.NET websites?

We are running Windows Server 2008 R2. TLS 1.0 has been non-PCI compliant for some time now, and disabling it via the windows registry is easy. In the past, though, disabling TLS 1.0 has caused 2 problems for us:

1. It's impossible to connect to the server via Remote Desktop.
2. Our IIS 6.1 websites become unavailable.

Problem #1 was fixed with MS patch KB3080079, but our websites still crash as soon as TLS 1.0 gets disabled. What do I need to do to keep ASP.NET happy, and the sites online, when disabling TLS 1.0?

Here is the error message, which only occurs when TLS 1.0 is disabled in the registry and after the server is rebooted:

Server Error in '/' Application.
The system cannot find the file specified
Description: An unhandled exception occurred during the execution of the current web request. Please review the stack trace for more information about the error and where it originated in the code. 

Exception Details: System.ComponentModel.Win32Exception: The system cannot find the file specified

Source Error: An unhandled exception was generated during the execution of the current web request. Information regarding the origin and location of the exception can be identified using the exception stack trace below.

Stack Trace: [Win32Exception (0x80004005): The system cannot find the file specified]

[SqlException (0x80131904): A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: Named Pipes Provider, error: 40 - Could not open a connection to SQL Server)]
   System.Data.SqlClient.SqlInternalConnection.OnError(SqlException exception, Boolean breakConnection, Action`1 wrapCloseInAction) +6749670
   System.Data.SqlClient.TdsParser.ThrowExceptionAndWarning(TdsParserStateObject stateObj, Boolean callerHasConnectionLock, Boolean asyncClose) +815
   System.Data.SqlClient.TdsParser.Connect(ServerInfo serverInfo, SqlInternalConnectionTds connHandler, Boolean ignoreSniOpenTimeout, Int64 timerExpire, Boolean encrypt, Boolean trustServerCert, Boolean integratedSecurity, Boolean withFailover) +6775368
   System.Data.SqlClient.SqlInternalConnectionTds.AttemptOneLogin(ServerInfo serverInfo, String newPassword, SecureString newSecurePassword, Boolean ignoreSniOpenTimeout, TimeoutTimer timeout, Boolean withFailover) +219
   System.Data.SqlClient.SqlInternalConnectionTds.LoginNoFailover(ServerInfo serverInfo, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance, SqlConnectionString connectionOptions, SqlCredential credential, TimeoutTimer timeout) +6777754
   System.Data.SqlClient.SqlInternalConnectionTds.OpenLoginEnlist(TimeoutTimer timeout, SqlConnectionString connectionOptions, SqlCredential credential, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance) +6778255
   System.Data.SqlClient.SqlInternalConnectionTds..ctor(DbConnectionPoolIdentity identity, SqlConnectionString connectionOptions, SqlCredential credential, Object providerInfo, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance, SqlConnectionString userConnectionOptions, SessionData reconnectSessionData) +878
   System.Data.SqlClient.SqlConnectionFactory.CreateConnection(DbConnectionOptions options, DbConnectionPoolKey poolKey, Object poolGroupProviderInfo, DbConnectionPool pool, DbConnection owningConnection, DbConnectionOptions userOptions) +1162
   System.Data.ProviderBase.DbConnectionFactory.CreatePooledConnection(DbConnectionPool pool, DbConnection owningObject, DbConnectionOptions options, DbConnectionPoolKey poolKey, DbConnectionOptions userOptions) +72
   System.Data.ProviderBase.DbConnectionPool.CreateObject(DbConnection owningObject, DbConnectionOptions userOptions, DbConnectionInternal oldConnection) +6781425
   System.Data.ProviderBase.DbConnectionPool.UserCreateRequest(DbConnection owningObject, DbConnectionOptions userOptions, DbConnectionInternal oldConnection) +103
   System.Data.ProviderBase.DbConnectionPool.TryGetConnection(DbConnection owningObject, UInt32 waitForMultipleObjectsTimeout, Boolean allowCreate, Boolean onlyOneCheckConnection, DbConnectionOptions userOptions, DbConnectionInternal& connection) +2105
   System.Data.ProviderBase.DbConnectionPool.TryGetConnection(DbConnection owningObject, TaskCompletionSource`1 retry, DbConnectionOptions userOptions, DbConnectionInternal& connection) +116
   System.Data.ProviderBase.DbConnectionFactory.TryGetConnection(DbConnection owningConnection, TaskCompletionSource`1 retry, DbConnectionOptions userOptions, DbConnectionInternal oldConnection, DbConnectionInternal& connection) +1089
   System.Data.ProviderBase.DbConnectionInternal.TryOpenConnectionInternal(DbConnection outerConnection, DbConnectionFactory connectionFactory, TaskCompletionSource`1 retry, DbConnectionOptions userOptions) +6785863
   System.Data.SqlClient.SqlConnection.TryOpenInner(TaskCompletionSource`1 retry) +233
   System.Data.SqlClient.SqlConnection.TryOpen(TaskCompletionSource`1 retry) +278
   System.Data.SqlClient.SqlConnection.Open() +239
   System.Data.Common.DbDataAdapter.FillInternal(DataSet dataset, DataTable[] datatables, Int32 startRecord, Int32 maxRecords, String srcTable, IDbCommand command, CommandBehavior behavior) +292
   System.Data.Common.DbDataAdapter.Fill(DataTable[] dataTables, Int32 startRecord, Int32 maxRecords, IDbCommand command, CommandBehavior behavior) +487
   System.Data.Common.DbDataAdapter.Fill(DataTable dataTable) +296
   FitTrack.Objects.Helper.Utilities.GetDataTableForQuery(String sql, Hashtable paramList, Boolean isStoredProc) +1002
   FitTrack.Objects.Helper.Utilities.GetDataTableForQuery(String sql, Hashtable paramList) +63
   FitTrack.Objects.Base.FitTrackPage.GetFitTrackPage(String pageName) +197
   FitTrack.Objects.Base.FitTrackBase.Authenticate(Boolean redirect, Boolean isloginpage, Boolean checkreg, Boolean isPayrollReport) +659
   FitTrack.Default.Page_PreInit(Object sender, EventArgs e) +68
   System.Web.UI.Page.PerformPreInit() +49
   System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint) +1844

Version Information: Microsoft .NET Framework Version:4.0.30319; ASP.NET Version:4.0.30319.34280

Note: TLS 1.1 and 1.2 were manually enabled in the registry.

We had to configure .NET to use stronger protocols before disabling TLS 1.0 to avoid System.InvalidOperationException in IIS.

REG ADD "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\.NETFramework\v2.0.50727" /v SchUseStrongCrypto /t REG_DWORD /d 1 /f
REG ADD "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\.NETFramework\v4.0.30319" /v SchUseStrongCrypto /t REG_DWORD /d 1 /f
REG ADD "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\.NETFramework\v2.0.50727" /v SchUseStrongCrypto /t REG_DWORD /d 1 /f
REG ADD "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\.NETFramework\v4.0.30319" /v SchUseStrongCrypto /t REG_DWORD /d 1 /f