Hello @Karl Swierenga ,
Is this the instrumentation key?
No, you have to create API key by yourself. The following documentation is a part of Bot Service, but it describes how to create API key and how to retrieve Application ID in Azure Portal.
If you are familiar with Azure CLI, check the following URL; Azure CLI reference to create API Key of Application Insights.
If you're using Azure Monitor connector to retrieve data from Application Insights, how about using managed identity authentication? It allows you to configure connector much easier. You should grant Reader or Monitoring Reader role of target Application Insights to a managed identity of Logic Apps.
https://learn.microsoft.com/en-us/connectors/azuremonitorlogs/ https://learn.microsoft.com/en-us/azure/azure-monitor/app/app-insights-azure-ad-api