The official name of this road is Yilan Route 1 宜一線. It's a great way to get you deep into the mountains for a scenic drive. The road quickly rises above the ocean and heads west toward Taipei county. It isn't very well known, so this is also a good way to avoid traffic and crowds.
The only challenge is finding the road. First you need to drive along the main road heading north of Toucheng to Daxi 大溪 . This is home to Honeymoon Bay, the surfing beach. After you pass the Daxi train station, you will eventually come to a bridge over a little river. Just before you get to the bridge is an old bronze bust of Chiang Kai-shek on a pedestal by the side of the road. Go up the road behind the bust. This passes through a little community of houses and quickly ascends into the mountains. You just keep driving along this road as long as you want. If you keep going, it will take you into Taipei county.
Above Daxi are some hiking trails which can be accessed from the road. If you're interested you can park and go for a walk. Otherwise there isn't anything in particular to do. There are a couple of Buddhist temples and farm houses along the way but not much else. The point is to just enjoy the beautiful mountain scenery and cool clean air.
After 40 minutes you will come to Hudiao Lake 虎貂潭. You can't see the lake from the road, but you will see a police station, then a white tiled building with a parking lot. You can park here and take a walk down to the lake and along the river. The walking trail is extremely well constructed - it's almost too good. The path seems like something you would find in a suburban garden. The riverside scenery here is extremely beautiful. You can swim in the shallower sections of the river, although swimming is (theoretically) prohibited in the deeper parts and the lake. Taiwanese assume that people will instantly drown if the water is more than knee deep. Unlike most swimming spots in the mountains, the water isn't freezing cold because it passes through a series of lakes that warm it up a bit. This makes it very comfortable for swimming.
If you keep driving about ten minutes beyond the lake area, you will come to a pavillion with a view of the sea along the north coast. From here you can keep going down into Taipei county, or else turn back.