Happy New Year.
I went to Taiwan thrice. Out of these trips, I went to æ·¡æ°´ twice. First time mainly at the old street near the MRT (popular tourist spot) and the second around the historic site and 渔人ç 头 (no old street, just went there to get some makan). I usually allocate a day for æ·¡æ°´, 北投,士林夜市 together cos they fall under the same MRT track, æ·¡æ°´ is the last stop. What I suggest is to set off in the morning at about 9am for hotspring 北投, then go to æ·¡æ°´ to spend the day and evening. Lastly back to 士林夜市 for dinner and shopping (they open til very late). But I think with the day filled with activities, you can be very tired by 10/11pm.
阿里山: I went there once and stay only a night. Saw sunrise only. The phenomenom is you either see sunrise or sea of clouds. Can't have them both on the same morning. Usu, if u get to see alot of stars at night, the next day should be a sunrise. Try staying for 2 nights, see if you see both. Anyway, there is posters to purchase, so no regret for me.
OIC. I can't recommend you the hotspring in 北投 cos I only went to one where u only soak ur legs (FOC). I went hotspring at YangMingShan.
Just a gentle reminder: For those require swimsuit, they oso insist you wear a swimming cap. If you dun have, u have to buy one before entering the place. The price is quite reasonable, but if you have it, bring it loh!