I've been to Taiwan twice, with another trip in planning.
English skills are generally weak (but occasionally excellent), but people are fairly friendly/helpful.
Accommodation: The windowless room thing is horrible, but at the same time a welcome way to avoid the considerable street noise in many places.
Night markets: I find Shilin overblown. Shida and other smaller ones are more accessible and enjoyable in my opinion. But don't limit yourself to night markets anyway... there's much more to Taiwanese food than that
I like Taipei, but not enough for 10 days. The older area north of Xinmending has interesting character. I find the area around Taipei 101 utterly soulless.
The National Palace Museum is great, but packed with Chinese and Korean tourists (and when I say packed... !!!). Lots of great stuff. Changing of the guard at Chiang Kai Shek Memorial is kinda fun.
Too many people don't venture southwards (on either coast). Taichung has a few charms and good markets, and a great modern art museum. Tainan was a bit underwhelming, despite its historical importance to Taiwanese. I haven't made it to Kaohsiung or Taitung yet. Hualien is tiny and has a great nightmarket. Taroko Gorge (nearby) is stunning and really should be done with a small tour (many paths are closed intermittently due to landslides, and the guide will know where to take you).
Sun Moon Lake is quite nice (but will be packed with tourists), but do make sure you cross or circle the lake, rather than hanging out on the built-up tourist-hub side. I don't recommend trying to do SML from Taipei.
Alishan mountain is apparently great (but will be packed with Chinese tourists). I don't recommend trying to do Alishan from Taipei.
Jiufen is incredibly busy. Pretty, though. But packed. You could stay somewhere in the northeast and visit from there. I wouldn't stay in Jiufen (I've done it!).
Carry some cash. Especially if away from a city centre or to pay small hotels or shops, even in Taipei. I've been in places where there was one ATM and it wasn't in the mood for foreign cards all weekend (just a glitch, not usually a problem), so all I could do for dinner was use the credit on my Easycard to buy snacks at a FamilyMart.
Train travel is good in Taiwan, though sometimes long distance coaches are more convenient due to the location of train stations vs bus stations. Booking a train can be a bit of an adventure for popular routes, and this blog was very helpful for me before my second trip:
Travel Taiwan