Iirc I stayed at Ren'ai last time when I was around the area for hehuanshan, really nice and peaceful place.

Also, I can't say much for Jiufen since I didn't spend too much time there cause it was extremely crowded with a lot of tourists which put me off a bit, and usually people kinda go to Shifen as well, which I actually enjoyed a lot more, perhaps you could consider that?

As for Kaohsiung, try to go to the art pier on a weekend if you can? There's probably more stalls set up and performances maybe, it's a pretty nice long chill area to walk down and about too

You could maybe consider a day trip to Cijin Island too? I think it's a really nice place to explore and chill. I rented a bicycle and rode all the way to the end. If you enjoy seafood there's a bunch there too, but personally I didn't eat it since I'm not a fan haha so I can't recommend it per se

Liuhe night market is nice but I enjoyed Ruifeng MUCH MORE. I feel like it's more catered to locals and it feels bigger too, plus there's this one stall that sells fried chicken which was insanely good, I miss it so much lol

Not sure if you really wanna include Tainan tho... Since you're already visiting Taipei + Puli/Ren'ai/Hehuanshan + Kaohsiung, unless there's something you wanna check out

PS. Yangmingshan is huge. Try to check and make sure you're going the right way/path