When trying to log into the backend Sitefinity logs the following error message in the log files:
 

Type : System.ArgumentException, mscorlib, Version=4.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e089
Message : The configured connection string SitefinityDiagnosticsModule does not exist in the DataConfig.
Source : Telerik.Sitefinity.Diagnostics
Help link :
ParamName :
Data : System.Collections.ListDictionaryInternal
TargetSite : Void InitializeConnectionSettings(System.Collections.Specialized.NameValueCollection)
HResult : -2147024809
Stack Trace :    at Telerik.Sitefinity.Diagnostics.Tools.DataAccess.DataAccessProfilingFactory.InitializeConnectionSettings(NameValueCollection config)

   at Telerik.Sitefinity.Diagnostics.Tools.DataAccess.DataAccessProfiler.Initialize(NameValueCollection config)

   at Telerik.Sitefinity.Diagnostics.DiagnosticsModule.InitializeTools()

The Diagnostics module has not been upgraded properly and when going to Administration -> Modules & Services there is an error message when clicking on the Details link saying "The configured connection string SitefinityDiagnosticsModule does not exist in the DataConfig."

The issue occurs if the Diagnostics module is enabled and the connection string for the Diagnostics module is moved to the web.config file and it is missing from the DataConfig.config file which is located under the ~\App_Data\Sitefinity\Configuration folder.

Define the connection string for the Diagnostics module in the DataConfig.config file located under the ~\App_Data\Sitefinity\Configuration folder. If the configurations are stored in the databse, check the DataConfig entry in the sf_xml_config_items table and add the connection string.

If the connection string is missing from both web.config and DataConfig.config file, disable the module as suggested below:

For Sitefinity versions 9.0 and above:
 

1. Execute the sql query against a backup of the Sitefinity database:

  

Update [YourDBNAme].[dbo].[sf_module_vrsn]
  set version = NULL, error_msg = ''
  where module_name = 'Diagnostics'

2. Restart the application
3. Run the project in browser

In order to install and use the module again after performing the above steps:
1. Go to Administration -> Modules and Services
2. Click on the Actions link of the Diagnostics module
3. Select Install
Sitefinity will initialize the Diagnostics module and will create the database for it and will add the connection string to the DataConfig.config file.