I checked Orange Hotel, location wise it's good, near Ximenting and still within walking distance to Taipei main station. Too bad that it doesn't offer 4-pax room.Few years back I stay in Orange Hotel - Ximen Taipei
Location to ximenting is around 3 mins walk toward the mrt exit1.
If you want cheaper, can check out Airbnb, usually have bigger room at lower price.
Airbnb I also checked. Ya there are more choices, just that most impose strict cancellation policy so we really have to be certain with our itinerary before we make booking.
Price wise, it's about $70 more per night than Cosmos (if you want room with window).