Hi Wendees,
I will recommend East Dragon Hotel at Ximending as I stayed in Rainbow hotel before and it is very noisy unless u intend to sleep at 12 midnight every night then it is fine. Dragon Gate is just 5mins walk away, 3 star also but very nice and clean room with hot water dispenser outside every floor. Book directly from their website, it is safer.
Hmmm, u want to make No2 in taiwan? U want to go hot spring? U take the MTR to beitou stop (about 1hr journey) and go to this spring hotel call shui3 mei3 (water beautiful) hotel. Can book a hot spring room for 2hrs nia and do you know what inside. Don't worry, it is not sleazy one and quite famous. When u reach the MTR, ask the station staff for direction, it is abt 5mins walk away.
Then after that, you take the MTR to Dan Shui (think 1 stop away only, very near), which is a small fishing village and quite nice to just walk around and eat some food. U may want to catch the sunset but must see what time hor, as when i went there, the sunset at 6.30pm!!!! So i missed it lor. Hee...