Can a baby get sick from a cold room?

I spent the night at my mother in laws house. She didn’t relay her heater was broken until we got there. We didn’t have a space heater or anything. Last night the room me and my 3 month old slept in was cold. I’d say around 66 degrees. She had a sweaty head when she woke up. I had layers on her. Now this morning she sounds stuffy. I’m quite frustrated because I’ve kept her in pretty good health.