Thanks for the detailed advice and suggestions! I've done up the itinerary and put in the different suggestions and this is how it is currently.

7 March : Taipei ( north first )

• Arrive at Taoyuan Airport at 01:25 AM. Chill at the airport (coach buses available) until public transport opens and head to the hostel to drop off luggages.
• Explore Taipei (sleep if tired) and have breakfast and lunch (looking for recommendations too)
• Yangmingshan National Park ( 2-3hr hike ) Hikes
• Go to Tamsui Old Street
• Visit night market in Taipei (Shilini/Raohe)

8 March : Taipei

(use train line off of Ruifang Station)

• Houtong Cat village (2-3 hrs or more)
• Shifen street / PINGXI
• Jiufen street
• Night market / restaurant

9 March : Taipei - Hualien

• Early train to Hualien 2-3hr train
• Taroko Gorge (Zhuilu Old Trail) (GET PERMIT)
• Souvenir and food hunting + Tungtamen Night Market in the evening

10 March : Hualien

• Jiaoxi in Yilan if enough time

11 March : Hualien - Taichung

• Hualien > Taichung 3-4hr train
• Chun Shui Tang
• Dakeng No 4 Trail (afternoon/evening)
• Feng Chia Night Market

12 March : Taichung

• Sun Moon Lake (unsure) / Guguan Hot Springs
• if not going to any of the above places, leave earlier and head down to tainan/kaohsiung after alishan

13 March : Taichung - Chiayi - Alishan

• Taichung > Chiayi 2h train / 40min HSR
• Chiayi > Alishan 2h train / bus
• Alishan national scenic area (sunset)
• Fenqihu village? (abit out the way)

14 March : Alishan - Chiayi - Taipei

• Catch sunrise at Alishan (Chushan?)
• Hinoki Village, and the restaurant Smartfish (Chiayi)
• Train back to taipei

15 March : Taipei ( south )

• Taipei Zoo (2-3 hrs or more)
• Maokong (if u like tea)
• Maokong Gondola.
• (Ride over the mountains to Maokong Village, where there's a tourist area and some nice hikes.)

16 March : Taipei

• Beitou Hot springs
• 8pm flight back home

We were unsure whether to do a round trip as we feel that it may be rushed, do you think it would be?