How do I install a sink in a room that doesn't have a drain vent?
Yours is a not uncommon problem, so a solution does exist. For example, how might one vent a sink in an island in the center of a kitchen? I looked online, and it seems there are two common solutions. You can use what is called a vacuum breaker, which lets air into the line to prevent slow/gurgly drain problems without allowing sewer gases into the area. The second option is an island loop vent. Both are described clearly in this link:

http://www.familyhandyman.com/DIY-Projects/Plumbing/Kitchen-Plumbing/how-to-plumb-an-island-sink