You will need to connect your HDD to your Mac Mini once you swap the hard drives. There are two options, a universal adapter which will allow you to connect the HDD and get the data or an enclosure, wich will do the same but at the same time could act as an external HDD as it protects the drive.
Now that you have a way to connect the old HDD, you will need to install OSX. To do so you will first need to format the SSD. Once you've done that you can install OSX and use the migration assistant to move all the data from your old HDD.
Hope you find this useful.
Here is the universal adapter.
By the way, here is the guide to replace your HDD with a SSD.
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