I woke up at 9 a.m. in the morning

(1) You either write "a.m." with both periods or "am" with none at all.

(2) It is more natural to move the phrase "in the morning" from the end of the sentence here. From my experience, the specific time and time of day generally come together: "9 a.m. in the morning" is treated as one phrase. If you want to separate them, then the time of the day will typically come first, as we often first start with the less specific information. However, this may still sound a little less natural.

> "I woke up at 9 a.m. in the morning." ✅
> "In the morning, I woke up at 9 a.m." 👍
> "At 9 a.m., I woke up in the morning." ❌

and I was very sleepy in the morning.

It is more natural to omit the second "I", as you have already used it at the start of the same sentence.

I really want to solve my sleep deprivation problem but I can't sleep early in theat nights, especially ion summer nights.

(1) A comma will typically precede "especially".
(2) The preposition to use with phrases like "summer nights" is "on" and not "in". Here's another example: "I like to have a hot beverage on winter nights."

After spending time with my momher I took a printout of the university exam which occurred last week.

I would replace "my mom" with "her" to avoid excessive use of "my mom". It is generally good practice to avoid explicitly repeating the subject too many times. The reader will understand who "her" is referring to.

And I started to solve eEnglish questions but my score was not that good.

(1) "English" is the name of a language, so it's a proper noun and should be capitalised.

(2) Starting a sentence with "and" is not wrong, but here's how I would write it:
"After spending time with her I took a printout of the university exam which occurred last week and started to solve English problems. However, my score was not that good."

I would prefer this, since writing "took a printout" and "started to solve" in the same sentence emphasises that they were things that occurred in succession and establishes an order of events.

I think I am a bit of a perfectionist.

After that I watched gGilmore gGirls.

"Gilmore Girls" is the name of a drama series and should be properly capitalised.