Father of two here, not a pediatrician. Both my kids (boys) were pretty easy sleepers within the first 6-9 months. It's hard to predict what will work, because every kid is different and needs vary. I have another on the way, so I'm crossing fingers that offering advice does not jinx it for the next one!
From my experience, I'd look at two variables:
First variable is attention, where she may just need to know you are "there". Our practice, whether you subscribe to it or not, was to eventually get the kids where they comfort themselves, up to a point. You say that she will fall asleep "on her own" but this is with contact and soft talking - just no holding. This isn't really "on her own", but I completely sympathize with your approach. With my first, we learned of the technique where you sit in the same room while they fall asleep (no other contact) and then slip out after they are good and out of it. Right now when she comes to, no patting means you are not around. Maybe you could get her used to you being there with no sound or patting so that when she wakes up, she doesn't need those forms of stimulation to feel comforted. If you subscribe to this, take a book or something, because it could take a while, and be sure not to acknowledge every movement or squirm, as that just encourages them to stay awake.
Another option, if if the first doesn't work, and since she is still so new to the world, is to bring the baby closer to you at night - you say she is on the monitor, so I suspect this means out of your bedroom. Try bringing the crib or bassinet in closer to you so she can hear you breathe or snore (as the case may be) and she doesn't have to subconsciously fret over where you are.
Second variable is to monitor daytime sleeping. It may sound counter-intuitive, but more or less sleep during the day may help her sleep at night. This depends on the kid. If mom and dad have arbitrarily determined the nap times in the day to fit their schedule, maybe let those go for a couple of days and see when and how long she sleeps on her own. Then make schedule adjustments accordingly. It will take a few days to get the rhythm right, but it could really help. Of course, if the baby is in day care, and you cannot adjust the nap schedule, this will be impossible, but may help on weekends.
If your baby does go to daycare, your future genius may be getting "overstimulated" from all of the activity and noise - her brain is processing all the info from the day at night. When she is home, try keeping things very simple. Keep sounds and activities quiet, soft and gentle for more than an hour before bedtime preparations start - longer if necessary. This way she has a bit of extra time to process and unwind from the day.
Hope something in this helps!