Probably the best place to stay on Alishan with a cost
Not in or even near Taipei! Very pretty area in the highest moutain range in south east asia. Alishan House (阿里山賓館) is an old Japanese era built lodge (around 1920's I think) getting a bit dated now, but I liked it. It was a favorite wtih the late Genralismo in the 60's & 70s. Some of the fixtures and furniture still seem to be from that era. Just be warned the rooms are smallish with no bar fridges and tiny bathrooms. Still it is the biggest and best in the area and a much larger new wing is being constructed behind it. The star gazing deck has fantastic views when the weather is clear.
Do NOT eat in the hotel, except for the breakfast buffet, you have been warned ... all the bad reviews I've seen on the net are true! The local village has a quite a few restaurants and are much better.