I would extend upon Sergey's answer too.
Тебе с ней ещё нянчиться придется. - You'll have to do more babysitting with her.
I disagree here. I would say that phrase conveys "start" rather than "more": "You'll have to start babysit her". The "more" motive might be implied here of course. Like, if you are babysiting 10 persons, adding 11th to the list is making more of the babysitting. Or, if you focus on time/effort spent, then starting babysitting makes you spend yet more time.
But those ideas are different.
You'll have to do more babysitting with her.
- you already babysit
- you babysit not enough, if you was babysitting for example for an hour a day, now you would babysit for three hours a day
Those implications are missing in "Тебе с ней ещё [потом] нянчиться придется".
What actually implied here is surprise, unexpected consequences.
You do not expect that your current decision would later add babysitting her to your current list of burdens, but it will. Think again if you want to make your burdens yet heavier than they already are. There would be a price to pay that you overlook and do not expect yet.
That is what conveyed here. "To all your chores, you would later have to babysit her as well", like this.
This notion of unexpected consequences, of a probably unpleasant surprise incoming is what conveyed by "ещё" here.
Еще и с ней puts emphasis on ней (her) which implies that you were babysitting someone else and now you will have to babysit her, too.
Ещё и нянчиться emphasises нянчиться
...and emphasizes it in the way, that you already has some list of activities you do with her, and now that list would get extended with babysitting.
Like, you were providing for her with money, food, medicine, and it is getting snowballing to the point it can now be called babysitting.
Or like you intend to do some limited activity, like going to parties together, but that person is known to call her acquaintances in the dead night and whine about her troubles, real or imagined. So, you get a forewarning, that you would have to BOTH walk her to the parties AND (unexpectedly) to wipe her night tears way too often.
Personally I derive it from how lists are described in Russian (not only in Russian, of course). Красный, желтый, синий И зелёный. Red, yellow, blue AND green.
In such phrases like these yours the inserted "И" implies some already existing list that would be yet extended. And the positioning of this "И" specifies which one of the few concurrent lists would get extended. Was it the list of activities with a certain persons, or the list of persons you engage in certain activity with or some other list.
So, the specific idea you want to convey is "her as well as him" - extending the list of persons for the specific activity. That positions "И" in front of "HER", thus "Тебе ещё и с ней нянчиться придётся".
Note, it can be worded without "ещё" - like "Тогда тебе и с ней придётся нянчиться". This would imply "It would not be that much of a surprise, and you probably already know it, but still you better consider it once again, for personally I would had not chosen this path".