Thanks to those of you who have kindly pointed out I forgot to include the most important detail, which DAY. This Sunday 6/23.
A couple of other details to mention: We will get to Cingtiangang around mid-day so a lot of water and sunscreen is advisable. We should be able to buy more water at Cingtiangang. It will be hot up there and for the first stretch of Fish Road, but as I remember most of it is under tree cover.
I haven't been to a ton of hot springs in Taiwan, but Baiyan is by far my favorite. It is totally natural and free. People have altered it, but I think the alterations are for the better. They've portioned it off into successively cooler pools and hacked a couple of stone chairs into the rocks.