I never thought that I'd enjoy Taiwan as much as I did when we first put the country on our itinerary. But to my pleasant surprise, it was actually a fun city to be in! I think 3 or 4 days would be enough time, because I was able to see everything I wanted to see during this time frame. The main attraction in Taipei (and on the List) is the National Palace Museum. I thought it was overrated but it does hold a lot of significance to the Taiwanese people. They say that there are so many artifacts on rotation, it would take a few months to see all of them. However, for me, one visit is a plenty.
It's also really close by to one of the hotels I stayed at during my visit, The Grand Hotel. Now, this hotel is a bit outdated but it's pretty magnificent to stay at. I suggest renting out the Chi Lin Room and not a standard room because you get better views and a much bigger room (including balcony). Plus the standard rooms aren't as well maintained as the Chi Lin Rooms and have super firm beds. The hotel also has an Olympic size pool and tennis courts that you can use. It is a bit out of the way from the city center, but there is a shuttle every 20 minutes to take you to their nearest train station. Another tip to know is that there is a free bus shuttle from the Taipei Main Station to the hotel, so you don't even need to spend money on a cab during your visit. But be mindful of your arrival because the free bus shuttle at the station stops at 6 pm each day.
If you want a swankier place closer to city life, another hotel we stayed at during my visit was the Shangri-La's Far Eastern Plaza hotel. Although this hotel isn't as visually stunning as the Grand Hotel, it does feel more luxurious. For only $60 more, we were upgraded to a larger room, obtained lounge access, complimentary laundry service up to $605 NT, and complimentary breakfast buffet. While there is an attached mall to the hotel, the shops weren't really that impressive. But hey, if you are bored, it doesn't hurt to explore it.
Other places I'd recommend seeing while you are in Taipei is the Chiang Kai-Shek Memorial Hall (like the Lincoln Memorial), Din Tai Fung (even though there are tons of them in the U.S.), Mango Cha Cha (their mango shaved ice is amazingly good), Chun Shui Tang (supposedly home to the original boba milk tea), and the Shilin Night Market (touristy but great for souvenir shopping).