Hi. I lived in Yilan (I-Lan) for about a year, so I have some recommendations.
Go to Zhengchang Xianrou Xiaolongbao (正常鮮肉小籠包)for some soup dumplings. It's located on Taishan Lu (泰山路) across the street from Guangfu Elementary school (光復過小). Simple operation, wonderful execution. They have kind of weird hours, though: 7:30am to 12:30pm and 3pm to 7pm. Be prepared to take a number and wait if it's busy (which it will be).
Also, try Du Xiao Yue 渡小月 for some Yilan food in an upscale atmosphere. From what I've heard, the chef is very famous throughout Taiwan. You can get some traditional Yilan foods here like gaozha 糕渣. I only had a chance to go one time, but all of the food I had there was really great. I especially liked the almond tofu they serve at the end of the meal. It's on Fuxing road, section 3, number 58 (復興路3段58號).
If you have a chance to go to Luodong (not sure if you're going to Yilan County or Yilan City), you should check out the night market there (it's much better than the one in Yilan). Mutton soup is very popular here. I think the most popular place is called A Zao Bo 阿灶伯.
Finally, if you REALLY like garlic, you should check out Beimen Suanwei Rougeng (北門蒜味肉羹). This is a thick porridge/stew made with slivers of meat, bamboo shoot, and tons of garlic. It's on Jiucheng Beilu (舊城北路) just west of Zhongshan Lu (中山路).
I hope this helps and reaches you before you go. I was a very poor student when I was living there, so I didn't get to try all of the nicer places that I would have liked, but I think you won't go wrong trying these recommendations. Also, I'm not sure what your level of Chinese is, so if you need help with directions or ordering or anything, just let me know.
I assume you can read Chinese, so if you go to http://nightmarket.funp.tw/, scroll down a little and look at the right side, they categorize the various links/reviews of night markets by region and name of night market. You will find Dong Men and Luo Dong night markets that supplement what Condimentality already said.
All I can say is, also try the Niu Si Bing (ox tongue pastry) that is a local delicacy and popular tourist/gift snack to buy. It's unlike the version going by the same name in Cantonese cooking that looks like a bloated Rolling Stones logo set of lips that's deep fried then sprinkled with sugar. This is like a flat multi layered cake, oblong shaped. Also can be very messy when eating, so be careful.
If you ever go to other parts in Yi-Lan county, Jiao Si is also very famous for hot spring spas and resorts, great for a getaway/relaxation. But what is even better than that, food wise, is if you go to some local seafood restaurant, you want to try goose, and more importantly vegetables like kung xin tsai (hollow stem leafy green) or even Chuan Chi (local thicker stemmed leafy green) as they are grown and irrigated with warm springs water. It's a flavor that's out of this world.
Yi-Lan also grows these gigantic scallions, supposed expensive by local standards, so if you get a dish that includes this in stir fry, give it a try. Very very tasty, and I'm inclined to think this is the equivalent of "tokyo negi" seen in Japanese supermarkets in the USA (like giant humongous leeks). A potential street food item you can try with it, is the Yi-Lan Tsong Bing, or the local version of Yi-Lan scallion pancake. It's a lot thicker than regular scallion pancake.
Yi-Lan is also famous for their kumquat, and there are a ton of delicacies, desserts, and dishes made with it.
re: K K
It's not a comprehensive listing of all the night markets (for example Gonguan night market's list of reviews is severely lacking), but it gives a glimpse as to what to expect.
This is actually a better travel guide for Yi-Lan county
and you can isolate by cuisine style or whatever leisure request category.
If you go to the main page www.travel-web.com.tw there is an English version of the page, but I bet it is not as detailed.
Treat it like a travel magazine. I suppose people pay to advertise and maybe you can print out online coupons for discount. Combine that with local blog search and you'll be able to plan your own trip.
re: K K
If anyone is looking for the Yilan scallion pancake, I've heard that there's a place on Minzu Street (民族路) that makes really good ones. I've never actually been there, but I've heard good things. From what I recall, it's either next door to Little Water Buffalo Shaved Ice (小水牛) or a couple store fronts away. The address for the shaved ice place is No. 385 Minzu Street.