"I like the mirror."

Translation: Ich mag den Spiegel.

[heba256812]

Why did we use " den" here?

[okakeiyou]

how do you identify when to use the 4 cases?

[az_p]

It requires a bit of time to understand until it becomes natural. I recommend looking for explanations on the internet that fit your learning style. Here is one that helped me (in a casual explanation style, but quite a long multi-part explanation): https://yourdailygerman.com/2013/12/06/german-cases-nominative-genitive/