Where I have been staying for over a week now on the east coast, Dulan village, Donghe township, Taitung county, the police station basically has a full camping setup intended for cyclists.
It includes several "bivouacs" which are basically well built low lean-tos with floors and roofs but no walls. You could sleep in those or probably set up a small tent in there. There'll be protection from rain as long as there's no typhoon.
There also free filtered water and a bathroom. The bathroom is even supplied with laundry soap power so you can hand wash clothes in the handbasin.
And I didn't realize it because there's no sign in English, but when I met a western couple cycling around the country, they told me you had to register inside the police station and you would be given the keys to the shower!
The couple told me it's like this all down the east coast. I never saw the bivouacs on the west coast but I also never thought of going inside to ask if there was a shower.
Now I had been taken to two or three police stations on the west coast weeks ago late at night by a couple of young local guys who didn't speak any English. None of the police offered me any amenities but the guys were only asking them about finding a park for me to camp in, so there is a possibility the police station on the west coast may provide some things and I just didn't ask the right questions or go to the right police stations.