I went Taipei near christmas. Unfortunately, it was raining heavily for my 4 days stay, before and after are sunny weather. Everyone was wearing winter coats and scarf, but we wore t-shirt and jeans that time, prepare a wind-breaker. Temp is abt 20 degC, consider air-conditioner temp. But up in mountain area is colder (10-15degC), you need to layer yourself. Taipei is colder than Kaoshiung as it is at the northern part of taiwan. But no need winter wear. This data is for those who really scare of cold weather.
Apart from those u mentioned, around Sun Yat-sen Memorial Hall, there is a street next to é¼Žæ³°ä¸° and near the famous dessert shop call Ice monster. The entire street sell nice, trendy clothing like those in Far East Plaza. I went there in Summer, hence not sure if they will sell winter wear during winter. Can also go to SOGO, it is a 7 storey high shopping department store. Enjoy the salesgirl and liftlady service. Very good. Another place is æ–°å…‰ä¸‰è¶Š. 101 Tower sell only branded clothing like D&C, Allure, Tommy, etc. Things and food there are expensive. I won't go shoopping during winter cos it is mainly winter wear not very practical for singaporean. And price is not cheaper for those in shopping centre, cheap stuff are mainly found in nightmarket. If you are going for comic and books, there is a book store street near Ximending. Comic are reasonably cheaper. If you sell many, they can give u discount. Since u r going to Shilin mkt, I'll suggest that you travel to æ·¡æ°´(Take a bus no çº¢14 to tour ard the town and the final stop is at é±¼äººç å¤´nice view during sunset)ï¼ŒåŒ—æŠ• in the day(hot spring), then can go shilin mkt at night. For these 3 places I mentioned, will need a day of walking around. Other than that, can go sightseeing and enjoy the hot spring and food at é˜³æ˜Žå±±.