Agreed. I would suggest one or two nights in Taroko Gorge. A day trip to Lugang and the rest of the time in Taipei.
Tainan is great but for a first time visitor, it is not worth the trip.
That’s plenty of time to visit Taiwan. My recommendation is skipping Taipei, and do 3 days on the east coast, and 3 days in the south.
Since it’s plenty cold in Germany, and you probably wouldn’t need more winter-like scenery, so perhaps skip Yilan, which is usually very cold, windy, rainy, and snowy on the mountains during winter.
So, maybe go straight to Hualien, then Taidong. Then head for Kenting (the beach town, which should still be pretty warm in winter for people visiting from Germany), Kaohsiung and Tainan.
Pearls to … 
Thank you, guys.
I have booked flights finally. I will land in Hong Kong on the morning of 19th December and go home from Taipei on the morning of 2nd January.
I will consider your ideas and maybe even do something else instead of Tainan.
Now trying to figure out how long I should stay in Hong Kong and when to fly to Taipei. Considering the 25th or 26nd of December for the flight to Taipei.