Today I asked for a (leave / a day off) to take my daughter to join thea school activity in the youth palace (?).

"leave" in terms of holidays is considered uncountable, so doesn't take an article.

You can also say "a day off" which is a little more casual, but also a little more frequently used.

"palace" is a big fancy historical building. Maybe your town does host activities for children in a palace, but I'm guessing you might have meant "youth place" here, which is more naturally written as "youth centre".

The teacher informed the students to arrive at 8:10 A.M.

I set the alarm at 7 A.M. (/ 7am) the night before.

Writing "7 A.M." with am spelled out as an abbreviation is valid, but it's more frequently used as just "7am" or "7 AM"

She dawdled putting on her clothes, so it took about 10 minutes.

Need a connecting word between those two clauses here

During that time I reminded her several times that she needed to hurry up, but she didn't care ito.

As a result we didn't set off until 7.:50 A.M.

Worse still, the car key battery was diead and I had to go back home to haveget a replacement, and the traffic this day iwas very very heavy.

"dead" is an adjective
"died" is the past tense of "die", so is a verb

If you wanted to use the verb here it would be "had died", but more natural to use the adjective I think.

At lastIn the end, we didn't arrive the youth palace until 8.20 A.M.

My daughter is so dawdling, thisdawdles so much, it really frustrates us.

Today I asked for a leave toso I could take my daughter to join the school activity inat the youth palacecenter.

I'm not sure what a youth palace is. If it's the name of the business, capitalize it to "Youth Palace"

The teacher informed the students to arrive at 8:10 A.M.

She dawdled while putting on her clothes, it tooktaking about 10 minutes.

Then she chatted while having breakfast and spent over 20 minutes on her breakfasteating.

The original correct, but this sounds slightly better.

During that time I constantly reminded her several times that she needed to hurry up, but she didn't care it.

You can also say "but she didn't care about it" or "but she didn't care about what I had to say". The addition of "constantly" instead of "several times" is only a suggestion, and helps with sentence flow.

Worse still, the car key battery was died and I had to go back home to haveget a replacement, and the traffic thisat day iwas very, very heavy.

You can also say "was dead", but "died" sounds more natural and implies it died on the way to the youth center.

At last we didn'tWe finally arrived at the youth palace untilcenter at 8.:20 A.M.

"At last" implies relief, which is correct and can be used here. However, I would pair it with the positive "At last we arrived at the youth palace".

I suggest using the word "finally" to convey a more frustrated tone of voice.

Today I asked for a leave to take my daughter to join thea school activity in thea youth palace.entertainment center.

“Youth palace” isn’t a common term in English. I assume it’s an entertainment center for youths.

The teacher informasked the students to arrive at 8:10 A.M.

After my wife made our breakfast, I woke up my (our?) daughter.

If she’s also your wife’s daughter, you could say “our daughter.”

She dawdled putting on her clothes, it tooktaking about 10 minutes.

During that time I reminded her several times that she needed to hurry up, but she didn't care it.listen.

As a result we didn't set off until 7.:50 A.M.

Worse still, the car key battery was diead and I had to go back home to haveget a replacement, and the traffic this oday is very, very heavy.

At lastIn the end, we didn't arrive at the youth palacecenter until 8.:20 A.M.

My daughter is so dawdling, thisdawdles so much that it really frustrates us.

The teacher informed the students that they need to arrive at 8:10 A.M.

During that time I reminded her several times that she needed to hurry up, but she didn't care about it.