[BUG] Deleting a Cluster Doesn't Reflect Unless I Reload

## Bug Description

Original Bug ID: 35826360

@ashu8912 was looking into this

The bug description was just the title.

Estimate: 2-3 weeks

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## Steps to Reproduce

1. Go to '...'
2. Click on '...'
3. See error

## Expected Behavior

A clear and concise description of what you expected to happen.

## Actual Behavior

A clear and concise description of what actually happened.

## Screenshots

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## Environment

- **Plugin Version**: [e.g. 0.1.0]
- **Headlamp Version**: [e.g. 0.12.0]
- **Kubernetes Version**: [e.g. 1.28.0]
- **Azure CLI Version**: [e.g. 2.53.0]
- **Browser**: [e.g. Chrome 118.0.0.0]
- **OS**: [e.g. macOS 14.0, Windows 11, Ubuntu 22.04]

## Additional Context

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## Logs

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## Checklist

- [ ] I have searched for existing issues
- [ ] I have provided all necessary information
- [ ] I have tested with the latest version
- [ ] I have included steps to reproduce

The issue provides no source files, tests, versions, concrete reproduction steps, or logs. Start by reproducing the cluster deletion flow in the AKS Desktop application and locate the code responsible for refreshing cluster state; done means the UI reflects deletion without a manual reload and the behavior is covered by an appropriate test.

Written by the indexing model from the issue text.