I did this road a few weeks ago...Huay Nam Dang - Sam Mun - Wiang Haeng.
Is it makeable on a streetbike? In the dry season most likely but it would be a slow pace and not enjoyable riding doing it, and a lot of work. You have to decide, is it worth it? you also have to be a rider who is confident and capable on a bike since you would be taking a streetbike out of its natural terrain. Its also not just 5 km's long either, its quite a bit longer. Its OK from the 1095 to Doi Chiang but as you get farther along it can get a bit rougher. In the current rainy season conditions I would not recommend it at all. It will be really slick especially on the Wiang haeng end of the trail.
Oh and don't take a wrong turn onto the singletrack trail, it would be bloody ugly.