You can't change the main connection string of Kentico database. It results in new database installation as you described. In your situation you will have to develop a custom code to connect to MySql database. Creating a custom webpart may be a good way. Basically what you need to do:
- Create a custom webpart - how to do it.
- Install MySql drivers - copying assemblies from MySql connector-net into your website bin should be enough. Package can be downloaded here.
- Connect to MySql database - tutorial here.
- Retrieve and display data in a way you want.