Lillian is super helpful and nice, she did help us a lot during our 5 day trip in Taipei, im sure u will definitely impressed by her hospitality.
The apartment is located in Dongmen Market, very closed to Exit 8 of metro, just the opposite of Yong Kang Street. There are shops and restaurants, breakfast, dessert nearby, very convenient of taking buses and metro to wherever u wanna go. For the transportation i would say Lillian's apartment would be one of ur best choices(specially travel with family).
For the condiction of the apartment, um... it's really small and i think it's better for 2 ppl max... we three traveled with 2 big suitcases and bought quite a lot clothes, have to spare space for our things everyday in order to keep everything in order and, the most important thing is to make spaces for moving inside the apartment. but the upper floor can contain 4-5ppl (3 mattresses).
a kind reminder: make sure u kill all the mosquitos the first night of ur stay, and if u do concern about if it's clean enough, i would suggest to bring a pillow case～
Overall, i would score 7.5 out of 10 for the place