My 2nd baby NEVER took a bottle. And I worked FULL time. That was tricky. Bless my babysitter and mother in law. Mom could get her to take breast milk with a straw -- Emily would kind of lick the straw. At four months, mom started giving her sips out of a cup. Not a sippy cup, just the cover to a bottle. It was thin enough, and just the right size for Mom to get a few ounces in her during the day.
So, if she won't take a bottle or sippy cup, maybe you could try a small regular cup. You will have to hold it, but it would let you be away from her when you have to be--the sitter can feed her.
As far as keeping her entertained, I think it is a good thing for them to have a little time on their tummies on the floor, on a brightly covered mat or blanket. Entertaining a baby all the time doesn't do her any favors really. My daughter that went to day care can't play independently like my daughter who stayed home with my mother in law. Tummy time helps them strengthen their upper muscles, something that is lacking in babies who are always on their backs.
And 15- 30 minutes of a musical DVD for babies isn't a terrible thing either! A happy mama is a happier baby!! So don't feel guilty using DVDs wisely. I loved the Richard Scarry ones, Little Bear, and Kipper. When they are older, Veggie Tales let me get dinner prepared every night!
Hang in there and enjoy every day!!