Set up an Amazon CloudWatch Logs metric filter to record each successful login as a metric, with the user name and client name configured as dimensions.

Modify the application's logic to trigger an AWS SDK call for each successful login, incrementing a custom CloudWatch metric with user name and client name dimensions.

Configure the CloudWatch agent to parse successful login metrics from the logs and additionally set up the agent to log these metrics as a custom metric in CloudWatch, using the user name and client name as dimensions.

Deploy an AWS Lambda function to process the application's log stream in Amazon CloudWatch Logs, and configure the Lambda function to increment a custom CloudWatch metric with user name and client name dimensions.