Install the CloudWatch agent server side and configure the agent to upload relevant logs to CloudWatch.

Enable AWS X-Ray tracing in API Gateway, modify the application to capture request segments, and upload those segments to X-Ray during each request.

Enable AWS X-Ray tracing in API Gateway, modify the application to capture request segments, and use the X-Ray daemon to upload segments to X-Ray.

Modify the on-premises application to send log information back to API Gateway with each request.

Modify the on-premises application to calculate and upload statistical data relevant to the API service requests to CloudWatch metrics.