HVAC blower control in several rooms
Ducts shouldn't be held (or even sealed) with tape. That's an obsolete technique. For a quick repair, you might find that aluminum foil tape is robust enough to hold. Otherwise, clean the duct surface and use clear 100% silicone. Give it at least 4 hours to cure. 8-12 would be better. Note that silicone isn't paintable, so if you need that you might consider urethane caulk. 

Really, an in-duct damper at the split to each room is best. You may need to get a tinbender involved, as special tools and techniques are sometimes required. 

Finally, consider switching the occupants of the rooms. Sometimes the best solution is the simplest one.