[Your English is extremely good. My edits are mostly just to make your sentences more 地道, not because they were actually grammatically incorrect.]

Depression

My favourite fast food restaurant is Burger King. Actually, though I don’t like the burgers there. The only reason why I always go there is that Burger King likes to play Eminem’s songs.

There was a period of time [unnecessary] when I went through suffered a severe depression ["to suffer a depression" is more common]. I never went out or got contact with contacted my friends [parallel structure sounds a bit better], and stayed all day at home, doing nothing. I only had a meal in a day then. I felt lost and alone. I was sinking like a stone. At Around that time [not quite specific enough for "At" to sound right], I began to listen to Eminem’s songs. My favourite song was Beautiful. I felt that [<- this may just be a personal preference] it described exactly what I exactly felt then ["what" is the thing that is "exact" here, I think, not "felt"].

Lately I’ve been hard to reach
I’ve been too long on my own
Everybody has a private world
Where they can be alone
..............
I just kept playing it again and again, [逗号s are more common in English than Chinese] and got a bit crazy. I thought I would stay depressed like this that [sounds more 地道，because your state has now passed] until the end of my life. However, Eminem’s experience of fighting against drug addiction aroused myself me to change the terrible situation I was standing in ["the situation I was/am in" is a very common saying]. So I decided to leave Nanjing and traveled [parallel structure] to another place to cheer myself up. On the journey, I kept listening to his another of his songs, Not Afraid. Fortunately, when I was back returned to Nanjing, I was made over into had become a new person.

Now I daren’t say that I’ve been fully out of my depression, but my situation is much better than before. I rarely play Eminem’s songs now. However, sometimes I would will [because you're stating something definite that you already do, in the present day] go to Burger King and listen to his songs quietly. It reminds me of that everything is going to be all right.

[This was good writing, a pleasure to read/edit. 加油！］

Depression

My favourite fast food restaurant is Burger King. Actually But I don’t actually/even like the burgers there. The only reason why I always go there is because that Burger King likes to plays Eminem’s songs.  

There was a period of in time when I was going went through a severe depression. (OR There was a time when I was going through a period of severe depression). I never went out or got in contact (touch) with my friends. Instead, I and stayed all day at home all day, doing nothing. I only ate had a meal in a day then. I felt lost and alone; I was sinking like a stone. It was at that time that, I began to listen to Eminem’s songs. My favourite song was 'Beautiful.' I felt that it described what I exactly how I was feeling felt then.

"Lately I’ve been hard to reach
I’ve been too long on my own
Everybody has a private world
Where they can be alone "
..............
I just kept playing it again and again, and got a bit crazy. (what do you mean by crazy?) I thought I would stay depressed like this until the end of my life. However, Eminem’s experience of fighting against drug addiction aroused myself inspired me to change the terrible situation that I was in standing. So I decided to leave Nanjing and traveled to another place (you can elaborate more on the place to make it more interesting. Was it a beach? A mountain? A village?) to cheer myself up. While on the journey, I kept listening to his another other song, "Not Afraid." Fortunately, when I was back in to Nanjing, I became was made over into a new person. (because of the song?or the place you went to?)

Now I daren’t cannot say that I’ve been  I am now fully out of my depression, but my situation is much better than before. (OR but I'm feeling much better than before). I rarely listen to play Eminem’s songs now. However, sometimes I would sometimes still go to Burger King and listen to his songs quietly. It They reminds me of that everything is going to be all right. (OR They give me hope that everything is going to be all right).

This note has a mix of 口语 and 书面语 if you get what I mean. For example, "however" is a little more formal than "but." You're not wrong or anything, it's just that this entry seems more personal so "but" might be better. And phrases like "got a bit crazy" is 口语. So the choice of words it depends on which direction you want to go. Good job! Keep writing =)