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Ah, the word "nanny"... it's just a word, but it means very different things in different parts of the world.
Having a "nanny" does not mean lots of money is on the scene, particularly if the dad is sharing a condo with a friend and splitting rent. It sounds like she's living on her own, with her own children. That tells me that she's more of a family friend who's taken in this kid because she might be friends with the family of the girl or her father, or might be from the same village, or a distant relative. That relationship is not always the purely financial one. Where was the mother living when she abandoned the girl? Was the mother living in another country with her daughter while her husband lived elsewhere? I might be reading a bit too much between the lines here, but I'm just getting a sense that this story has elements to it that have less to do with fabulous riches and more to do with cross cultural confusion. I have a good friend who was effectively raised by a "nanny" that was a distant cousin. Very little actual money changed hands in that upbringing, but lots of family ties and goodwill did.
I'm not going to get into the moral judgements of you or your partner, or the situation. It sounds like the nanny is effectively this girl's nurturer. The nanny is the constant, not you, not your boyfriend. Your guy isn't able or willing to nurture her in the way it seems she needs. I'm sure he thinks he's done an OK job with nurturing her in the way he probably feels he can, but it clearly isn't what she needs. The soiled towels and general helplessness isn't down to the nanny, it's down to the girl asking for the kind of attention and care she did not receive as a child. That being said, if the nanny is happy to have her, why not let her stay with the nanny? You both need to spend more time around this girl, but if you aren't cut out to nurture this girl, I don't see why either of you should pretend. You'll do a halfway job of it, you'll both be miserable, and she won't get what she needs from you.
This does tell you lots about how this man considers his responsibilities as a father and a carer. It sounds like he's moving the girl in with him now that he's living with you (a woman he'll possibly marry), and once she reached 16, an age of quasi-independence. But not before. I don't know if his parents are still living, but you need to think about how he'll ask you to be involved with them as well as with this girl.
If you decide to stay with this man, you will not be able to count on much beyond financial support from this man with your own children. If you are OK with that, and you're OK with the way he has/hasn't parented this 16 yr old girl, then stay. If not, then go.