To ensure that no sessions are lost if an Amazon EC2 instance fails, using Amazon DynamoDB to perform scalable session handling is the most effective solution. DynamoDB is designed for scalability, reliability, and low-latency access, which makes it an ideal choice for storing session data. By storing session data in DynamoDB, the data can be quickly retrieved and is independent of the lifecycle of the EC2 instances, ensuring fault tolerance and resilience in the system.