I've taken my girls on long trips to SC and MI since they were infants. My son made his first trip to SC (16 hour drive) when he was just three months old. It is harder when they start to get active, but there are things you can do.
1.) Get a dvd player for the car if she likes Elmo that is a great one. Get about four or five differnt kinds of dvds....Tom and Jerry, Elmo, Barney....all good ones for that age range.
2.) Get lots of snacks for the trip that she really enjoys.
3.) Stop at rest areas and let her run around about every three hours or so...you can go longer if she's doing well. Stop at McDonald's or some place like that once to let her play and stretch her legs.
4.) Go very early in the morning or even consider traveling at night while she sleeps if you and your hubby can stay awake.
Honestly if you don't start making the trip now then she will never be able to do it. Will you never visit family again? Connecting with your family and helping her develop a relationship with them is more important that your stress levels. Traveling is always a little stressful, but if you make plenty of stops, take snacks and so on...it will be fun. Take a book on dvd for you and the hubby (maybe two or three).
If you don't think she can handle 11 or more hours on the road then schedule a stop at a hotel with an indoor pool. Make it a fun thing.
Believe me I grew up on the road. My father's family lived 13 hours from us and my mom's was 3 hours away. We visited family at least two times a month. We went to MI atleast once a year.
She will be fine.
And who cares if she eats...it's a short vacation. My kids eat very healthy at home, but when we go visit family their diet goes into the toilet. Who cares...it is only for a few days. She will eat when she gets hungry.
And I understand how you feel about being the only one she will go to. My girls were the same and my son is there now. He's five months old and almost 19 pounds. I would love it if he would go to someone else, but he won't. We visit family every Sunday after church. I spend the whole day with him...but family and those connections are what is most important.