Sorry I didn't see this earlier. I personally have not returned to Taipei since 2009, but I still keep an eye out on random blogs and publications.
For a breakfast place, the one to try is apparently Fu Hang Dou Jiang 阜杭豆漿 despite the fact this is now on practically every Chinese speaking country's radar (HK, China especially, Singapore, Malaysia).
Be warned though, the wait is ridiculous at times, and now with the advent of very unruly "強國人", the wait could even be worse as some of them buy upwards of 20 shao bings at once for their friends/families (each oven roaster can only make a small batch at a time). Though if you are jet lagged and up at 5 am, get yourself to Shandao Temple station and brave it for some of the best in the country (or county) and hopefully before the line gets too long. If you love Shao Bing, this place might just be it, unless there's a new place in town that is just as famous and ridiculous.
For night market, my personal favorite was always Ningxia Road 寧夏夜市 as this place represented the heart of old Taipei with their old school recipes. Saw some recent Instagram pictures by friends who are there now and still looks excellent. Worth entering 寧夏夜市 in google search and see what the local bloggers suggest are the latest and greatest.