I am a mid-westerner who lived in Taiwan for a few years, and the most common way to get home is the NWA flight from Taipei to Osaka, then a long flight to Detroit, and then another short flight to Madison, or any other city in Wisconsin.
I took this flight many times, and it is usually the cheapest. You can also search on EVA, Malaysia Air and Singapore air, which will usually give you good prices from Taipei to San Francisco or LAX, but after that it is pricey to fly to the midwest.
I would choose the Northwest Airline.
I have always used them by choice hands down.
I have a sister that travels regularly for work and she has been on many different airlines...she has said nothing good about any other than Northwest too. They are reasonably priced as well. Never had any ssues with loosing bags or anything like with some of the other ones.
You should be able fly Northwest from Taipei to either Osaka or Tokyo. Then you arrive at Detroit and then catch your connection to Madison City. Northwest Airlines flies to many destinations in the upper midwestern states such as Wisconsin and Michigan.
either american (although they get very hard to find direct) or pacific cathay other than that search some ood booking sites that include lots of airlines and then choose from what they give you.
Eva or American airlines
try this site.......also book on the weekday - it is cheaper than weekends