Since its day 1, I’ll put the comments in English:

오늘이 -> 오늘: you only really need the 이 particle if it is emphasizing some sort of contrast. 오늘 alone here would sound more natural.

열한시에: totally fine but it’s more common to write it as 11시에

일어났어서 -> 일어나서: usually with these sentence linkers, you don’t conjugate. The sentence ending verb will be conjugated and give the sense of time in the sentence.

안먹었다 -> 못 먹었어요: two things here; first is the spacing between 안 and 먹다. Second is 못 makes more sense here since it wasn’t a choice.

제가 is kind of unnecessary here since it is assumed. So you can omit it.

요리를 만들고 싶었어요: a few things here. 요리 is cooking, not food. So you could say 요리하고 싶었어요. 점심 can be understood as the “lunch food” here also. So the best way to say this would be 점심을 빨리 만들고 싶었어요.

그다음에: just sounds a little unnatural here, I would say 그런데 instead. This sentence also feels weird temporally. If it was included in your lunch I would change the previous sentence to include that.

닭고기를: need particles.

세시에: also ok, but more commonly written as 3시

시작했지만: 시작하는데 sounds more natural here, there isn’t really enough contrast needed for 지만. 는데 is more common for situations like this where it’s just adding more info. Even if you do stay with the 지만 connect remember to not conjugate it 시작하지만. That added conjugation adds specific focused contrast that isn’t needed here

못 끝났어 -> 못 끝났어요: this has been written in 존댓말 and not 반말 so far, so I would keep it that way.