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

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

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

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

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

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

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

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

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

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

After my wife made our breakfast, I woke up my (our?) daughter. 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 while putting on her clothes, it tooktaking about 10 minutes. She dawdled while putting on her clothes, taking about 10 minutes.

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

Need a connecting word between those two clauses here

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

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

The original correct, but this sounds slightly better.

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

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

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.

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

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

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

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

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. Worse still, the car key battery died and I had to go back home to get a replacement, and the traffic that day was 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.

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. Worse still, the car key battery was dead and I had to go back home to get a replacement, and the traffic this day was 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.

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. Worse still, the car key battery was dead and I had to go back home to get a replacement, and the traffic today is very, very heavy.

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

"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.

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

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

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

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