With engorgement, perhaps you can try some cold compress to help with the swelling and pain (temporary relief) OR hot compress to get the milk flowing and then try and squeeze out some or pump out some to soften the swollen areolas - she can't latch on properly when you're engorged because the nipples will "disappear" beneath the areolas.
As for the latch-on, there are some things to look out for...
1) For the traditional hold, tummy to tummy contact is important - her tummy needs to face you and from her chin to her legs, she needs to form a straight 180° line. dont't let her body dangle away from you. If possible, keep her as close to you as you can.
2) When she's rooting, wait (patiently) for the moment when she opens her mouth super wide and then pull her in (not push your breast to her). You can help by using your forefinger and apply gentle pressure on her lower lip (area in between lower lip and chin) to open her mouth even wider. Her lower lip should fan out over your areola. The initial sucking can be uncomfortable because she can be vigorous but otherwise, you shouldn't feel any pain like a minute into sucking.
3) Her ears/earlobes would be wriggling as she sucks if your latch on is correct. If it's wrong, you'll see her suck in her cheeks instead of her moving her lower jaw.
If all this is still correct, chances are you're going through transitional tenderness. I would advise to continue nursing, apply some lanolin (PureLan 100 by Medela is god-sent), air your nips regularly and put on some nursing bras to give your nipples a break from the chafing.