I have multiple advanced degrees and accomplishments in life that leave people routinely asking, "How do you do it?" I also have twins. Let me be very clear: EVERYTHING THEY THINK IS IMPRESSIVE IS NOTHING COMPARED TO HOW HARD NEWBORN TWINS WERE.
Absolutely nothing in my life has compared to how hard it is to be home with two infants. And that's even with my church providing our dinners for two solid months. You get barely any sleep. You spend every waking moment changing diapers, feeding, and cleaning spit-up off your clothes and furniture. You might spend hours each day walking up and down with fussy babies when you're exhausted yourself and haven't had a decent meal in two days because there's no time to make one, let alone sit down and eat it. Sometimes, it's impossible to care for both at the same time, and you listen to one baby cry while you're practically in tears yourself trying to settle the other one. Your house is often a mess. You might not have clean clothes because you don't have time to get to the laundry. I could go on and on.
If my husband had left me 2-3 nights per week to be with a girlfriend when they were brand new, he'd be singing soprano. Not that it would matter, because I'd be through with him. (Actually--come to think of it, he spent a great deal of that time having coffee with a female friend while I was home alone dealing with two babies, and I AM through with him. We're now divorced.)
As a secondary myself, I'm normally perhaps a little bit touchy about remembering these outside relationships are human beings, too, and not disposable, but darn, she needs to grow up and realize you've got TWO new babies at home and a wife who needs help.
YOU need to realize that you brought two brand new human beings into this world, and you DO have obligations to them and to their mother/your wife.
Honestly, maybe now is not the time to have a girlfriend at all, unless she values you enough to value your children, too, and help out instead of pulling you away from them.