I would recommend to stay at the Landis Taipei. It's a short walk to an MRT station and pretty central. It's a 4 star hotel and reasonable priced. The staff was helpful and spoke good English. Clean and modern decor. With regards to things to do; How many days are you planning to spend there? Are you only going to be in Taipei or do you have time to go outside the city? You should check out the tea houses and old streets in Jiufan, and seafood/night market in Keelung. You can even go out to Yilan for a day which has a lot of greenery. If you have more than a few days to spare, I suggest taking the train down to Hualien or Taitung. The contrast between mountain and sea is beautiful and it's easy to rent a car once you are there, barely any cars on the road, very relaxing drive. If you want to explore other cities you also have the option of taking the high speed train to Taichung or Kaohsiung. This can easily be arranged as a day or overnight trip. Do also check out the Taipei Zoo, ride the Maokong gondola and check out the view of Taipei at the top of the mountain.
I stayed in Seeyou hotel the last time I was there. But that was a couple of years back. Found the location good, room was clean. Relatively easy to get to different places in Taipei via their metro. http://taipei.sunrise-business-hotel.com.tw/
Try cityinn at fuxing!! It's reasonably priced with washing machines so Toh can do some laundry before traveling home. It's conveniently located just few minutes from the train station and there's a lot of eatery just below the hotel.