I have some recent experience so I will share what I know.
I know of one place in Danshui that rents but they are wary of foreigners. My friend went with his international licence and ARC but they wouldn’t rent to him. I had to rent it for him because I have a Taiwanese bike licence. From what I remember the price was 400NT for a half day and the bike was a very old Kymco Jockey. Fine for one person but not much fun for two people.
IMO bike rental in Taipei and the north is a pretty poor experience. Poor choice, old and poorly maintained. There is a place at Taipei Main Station (may or may not help) that rents slightly newer Yamaha scooters. They are decent enough but lights are always on for oil change and the two scooters I have rented, while very stable being Yamahas, had rear brakes that were on their last legs.
This is in comparison to Kaohsiung, Hualien and Taichung where I’ve rented a nearly new Kymco Racing 150, a Yamaha BWS and a PGO Tigra with ABS.
There is a foreigner who rents scooters but collection and drop off is at Taipower Building MRT. I rented a motorcycle and it was fine. He also rents scooters and he seems to keep his stock safe well maintained.
Sorry I can’t be more use for Danshui and Jeelung.