Depending on the architecture and setup of Splunk in your org, you'll have to figure out what things needs to be monitored as part of the health check.
Apart from the basic, infrastructure aspects where Splunk is running, you'll also need to monitor the various components that builds the Splunk ecosystem. The Splunk specific things will be covered as part of the Monitoring Console component. Configure appropriate alerts, dashboards, peers, etc as per your org setup.
Start with the infra basics from CPU/Memory/Storage/Network and build your way up and tune those health checks for your Splunk environment.
Note: Depending on how the team currently monitors health, you can configure email notifications, other API based notifications and even use some third party applications for APM & Infra monitoring.
For more in-depth view on this, I would recommend reading the following documentation from Splunk.
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