(Updated June 2018) The Bayan Hot Springs are a gorgeous series of waterfalls and pools hidden within the Yangmingshan forest. They are one of the few 'wild' hot springs in north Taiwan. They are not covered in pipes, or organised into public facilities. It's just the springs, and the forest. There's only one other wild hot spring I know of on Yangmingshan, but that one is really unknown, and I think it's best kept that way.
The Bayan springs are fairly popular, and easy to get to, so on weekends they can be quite busy. I didn't find this a problem, as the place is so relaxing the crowds weren't a bother. There were also quite a few other foreigners there. This meant I could relax and leave them to answer the "questions" that can make other public hot springs a bit tiresome. Going on a weekday there were only another 8 to 10 people there.
The springs are technically illegal to enter, but they are very popular. I think it is purely to cover the administration in case someone injures themselves. We entered from the top and didn't see the sign warning of the fine as that is by the south entrance. As of 2018 I've been told there is more fining going on, though not for the full amount. A lady I was speaking to at the springs said the officials usually come in the middle of the day and that she just runs further up the river and hides until they are gone. Sometime in 2017 there was a flood and landslide which took out the small waterfall. However the springs are fine and still pretty.
(Update 2018: The route described below is closed. You must take the trail that follows the orange river upstream next to the Bayan Hot Springs Hotel. At the small stream crossing head uphill until you reach the springs)
From the Jinshan Road stop at the Bayan bus stop (link here). The infrequent 1717 bus from Taipei Main stops here. We scootered, but friends who took the bus said it was not as long a journey as Google makes it out to be. (I've since taken the bus too and it's not a bad journey).
From the bus stop head downhill through a series of pretty rice paddies and old farmhouses.
Eventually the path will turn to the left and into a forested area. Soon you will reach a barren sulfur vent field. Sadly there is a lot of garbage in the vents, I guess from people trying to see if it was hot enough to burn it.
Continue downhill and you will loop around the vent field and head downhill from it. At some point you will hear a small stream bubbling out of the vent areas. This is the spring which feeds the hot springs. When you reach a metal fence there are two options. We knew of no other way, so we walked down on rocks over the hot stream. Someone has put rocks and sandbags here to make it easier, but if you slip you would certainly be scolded. If you continue down the main trail you'll come to a turn off on the right which will lead past a warning sign and into the springs. This is much easier.
In the spring valley the hot stream mixes with a cold water flow entering the valley via a small waterfall. This makes it comfortable to bathe in. There are a series of pools, each mixed a little bit more with the cold water. The top pool was way too hot for me. Next to the hot spring pools the cold stream cascades over some rocks and pools, and makes a good place to cool off. The springs aren't totally wild, people have built small steps and pools, but it still looks beautiful.
I've marked out the trail on the Hidden Taiwan Map so you can see roughly how to reach it. This is well worth a special day out to and I look forward to returning.