In addition to the other corrections already made:

I washed my face and drank a coffee with a sandwich.

Then I woke up my daughter OR Then I woke my daughter up.  Both sound correct to me.

We put on clothes. OR:  We put on some clothes.  Both sound correct to me.

After the kindergarten I drove to work.   The word "to" is necessary.

At midday I went to a canteen [Amer.Eng. - cafeteria] for lunch.

After work, I picked up my daughter from the kindergarten and we drove to home.

sport section?? I'm not sure what "sport section" means.  

We ate the supper, and talked about our day, and watched TV. Then we took a showers and went to bed. And that's how it is everyday, just like in the movie "Groundhog Day" :)

My yesterday

I write about my yesterday. I got up early as usual. I washed my face and had a coffee with a sandwich. Then I woke my daughter up. She washed her face, and I braided her hair. We got dressed and I drove her to the kindergarten. After dropping her there, I drove to work. I worked hard all day as I always do. At midday I went to a canteen for lunch. After work I picked up my daughter from the kindergarten and we drove to home. At home I made a supper and drove my daughter to sport section* - ice skating. While Kira (my daughter) was engaged, I was reading a book. After ice skating we drove to home and were waited for my husband to get back from his work. We had the supper and talked about our day, after which we watched TV. Then we took a shower and went to bed. And so everyday as in the movie "Groundhog Day"* :)

*Don't know what to do with that.

My yesterday

I am writing about my yesterday. I got up early as usual. I washed my face and drunk a coffee with a sandwich. Then I woke my daughter up . She washed her face and I braided her hair. We put on some clothes and drove to the kindergarten. After the kindergarten I drove to work. I worked hard all as always in last time. At midday I went to a canteen for lunch. After work, I picked my daughter up  from the kindergarten and we drove to home. At home I prepared a supper and drove my daughter to sport section - ice skating. While Kira (my daughter) was engaged there , I was reading a book. After ice skating we drove to home and we were waiting for my husband to get home from his work. We ate the supper and talked about our day, we watched TV afterwards. Then we took a shower and went to bed. And so everyday as in the movie "Groundhog Day" :)

Well written!!
just be careful with phrasal verbs like : pick up. .
When you use this you need to remeber this: pick (something or someone) up.. so you say I picked my daughter up  instead of I picked up my daughter.. that's why I highlighted this..
And you said engaged.. this is used when someone is dating.. and then they get engaged.. and then they get married.. allright?
Keep up with the good work