Hi banboke,

This is a very good entry. I am impressed by your English skills. Here are the corrections I would make: 

My husband and I went to Taiwan from September 1st to September 5th.

My husband and I went to Taiwan from September 1st to September 5th.
It was a 3-night, 5-day trip. 
My parents-in-law picked us up at 8:00 o’clock at night. (8:00 is read as 8 o'clock)
We left Japan at 22:45 at night and arrived at Taiwan Taoyuan airport at 1:00 a.m. on September 2nd.
We checked into the hotel at 2:30. It was hot and humid.
I was very tired.

On the second day, we checked out at 6:30 when a staff at the hotel weren’t there. ("On the second day" is not a full sentence. In English we would not end this with a period. "Wasn't" is singular. Since you are talking about multiple staff, we would use "weren't," which is the plural version). I put a room key at the key box.
I forgot to ask a staff to keep my luggage until in the evening. (Staff is actually plural, so we do not use "a" before this word in English)
So, we had to look for a coin locker around the Taipei station. It took about 40 minutes to find it . My husband got mad at me .
We really wasted of time.
Our breakfast was a butter bread and a glass of tea in the convenience store . At first we went to Yeliu Geopark .
From the entrance to the tope of the cape is 1.7km, width is 300 meters. It was a very dynamic strange erode rocks. (I do not understand this, but one option would be to change it to "strangely eroded rocks"). 
It was sunny and a good weather.
The rocks have been badly eroded by wind and rain.
I can see a beautiful ocean and a brown rocks. It was really fantastic.
Next we went to Wurai where a Tayal tribe was lived. 　There was a hot spring.
We visited a museum.  I learned that they had a habit of tattooing a face and headhunting, which is the practice of taking and preserving a person's head after killing the person because of religious custom.
I was really surprised.
The town was affected by the strong typhoon last year. (I am very happy when people use this verb correctly. It is tricky. Good work.)
A trolley was suspended . I was very disappointed.
I went to the Chayi at 20:00.
I had a dinner at a night market. I ate a grilled beef steak.
It was tasty.