i called and texted your friend this afternoon while you were at 故宮 3rd floor here are some information about the night markets
(1) 通化街夜市 - very close to 東區, where you are staying. you can take the brown line MRT to "六張犁" station and walk about 15 min on 基隆路 and you will be there. this night market is very balanced with food, clothes and shoes. 推薦食物: 肉燥飯 (at the middle of the night market), 香腸 (Also around the middle section of the night market ((karen likes the garlic flavor))), and 生煎包 at the beginning of the street on 基隆路's end.
(2) 饒河街夜市 - it's "藥燉排骨" is the best in taipei area (maybe the whole Taiwan). you can take the blue line MRT to 市政府 station and change to taxi (it should take less than 10 min to get there from the station so the fee should be less than 100 before 11pm.) note: there are many stores that sell 藥燉排骨, but the one i recommend is the one in the very middle and center of the night market called 陳董藥燉排骨.
(3) 士林夜市 - (notice this night market's name doesnt have "街" in it.) this might be one of the biggest and best night market in Taiwan. the area it covers is so big it was not until recently i am finally able to walk around there without getting lost. to get to this night market you can take the red line MRT to 劍潭 station. the night market will be on your left hand side when you walk down exit #1. After you exit the station and face to the left, you will be able to cross the street but there is 2 parts of the night market; <1> on your left you will see parking lots and a bit further is a building where most of the good food are located. ( recommend: 大餅包小餅, 蚵仔煎, 牛排, 紅油炒手).
<2> on your right is the shopping area (clothes, shoes, toys...etc.), but it also have a lot of food inside. (recommended food in this area : 臭豆腐, 涼麵, 油飯.) this area is the part where a lot of people get lost due to its many branch out ally.
(4) 師大夜市 - this one is also close to 東區. you can take the blue line MRT to 台電大樓站 exit #3. after you leave the exit turn right and right again on the first corner. walk a few blocks down and you will see night market on your right hand side. recommend food: 牛魔王牛排館, 生煎包, 檸檬愛玉(喝的), 燈籠滷味 <---very famous.
Very sorry i wasnt able to help much during your stay in Taiwan this time. i hope this information reaches your hand and still allows you to visit some of these places. night markets usually start around 5~6 pm ends around 1~2am depends on the time of the week, weather, and business of the day. if you need anything or a last min guide around taipei dont hesitate to give me a call.
P.S: have you been to 鼎泰豐 yet?
如果是你們朋友第一次來台北玩 你會如何推薦台北的夜生活呢 (請不要說夜店 對方是家長)