Another good one quite nearby would be this: http://www.hel.fi/hki/Museo/en/Etusivu
The exhibition place - an old Russian villa - is situated quite comfortably 500 meters north from Kiasma along the Mannerheim street (Mannerheimintie in finnish). You can see Mr. Mannerheim himself near the entrance of Kiasma riding his bronze horse to the south, so you walk pass him. On the left side stands the monumental parliament house and on your right side they are constructing a new music house. You'll find the exhibition immediately behind the new music house construction site.