Hi,
I am looking for a Mandarin speaking preschool in Taipei, preferably near Shilin. I am hoping to find one that my 3, 5 and 6 year old can attend this summer while we are visiting Taipei. Alternatively, are there any good Mandarin speaking (local) one week camps (LEGO, sports, etc) for my 6 year old. Any recommendations would be greatly appreciated.
Mandarin being the official language in Taiwan, I don’t think you would have too many problems finding a Mandarin speaking nursery ! 
Whether you could find a nursery which accepts kids on such a short-term basis, is a different matter. So a camp may be your best option.
Is it only for a week, or are you just considering another option in case you can’t find anything suitable?
Places like Y17, Happy Bear, Mandarin Training Center all run summer programs, as does the science museum. I have introductions to all these places on my blog (http://www.kidzone-tw.com) in English, with links to their mostly Chinese information. Look under Indoor playspaces and Summer Classes. The problem with some of these (like Y17 and Mandarin Training Center) is it can be hard to get places in their programs, as they fill up fast. These places are not in Shilin but they are on the MRT line that goes through there. I also know of some other classes your kids could probably do together, run by a wonderful woman who also teaches Mandarin to several children I know here. I will find her information and pass it on to you. If you are looking for exposure to Mandarin during summer holidays and can travel around, you could also try all the library storyhours, events at Huashan or Dongshan Riverside Park’s Children’s Folklore Festival.
There should be preschool options, too, and you could try asking at http://www.parentpages.net if you don’t get enough information here.