Update LLM Router README.md for improved clarity and deployment instructions

- Revised the overview and table of contents for better organization.
- Enhanced quickstart section with detailed environment variable setup and deployment steps.
- Updated routing strategies and API usage examples for clarity.
- Adjusted version numbers and image references to reflect the latest updates.
- Removed outdated sections and ensured consistency throughout the document.

Author: arunraman

examples/README.md

@@ -36,7 +36,7 @@ Platform-specific deployment guides for production environments:
 
 - **[Amazon EKS](deployments/EKS/)** - Deploy Dynamo on Amazon Elastic Kubernetes Service
 - **[Azure AKS](deployments/AKS/)** - Deploy Dynamo on Azure Kubernetes Service
-- **[LLM Router](deployments/LLM%20Router/)** - Intelligent LLM request routing with NVIDIA Dynamo integration
+- **[LLM Router](deployments/LLM%20Router/README.md)** - Intelligent LLM request routing with NVIDIA Dynamo integration
 - **[Router Standalone](deployments/router_standalone/)** - Standalone router deployment patterns
 - **Amazon ECS** - _Coming soon_
 - **Google GKE** - _Coming soon_