I like the suggestions you have already! Here's my two cents' worth:
The way to get her to play independently is, first, to tell her that it's time she entertained herself. But then you have to help her learn how to do that. It's a lot different than turning on the TV.
You might talk to her teacher for suggestions. Teachers sometimes have a chance to notice things about a child that the parents don't.
Then, if your daughter likes art projects, get her some of that clay that won't dry out, a table knife, and a work area that the clay won't ruin, and say, "See how many things you can do with this in half an hour." It doesn't have to be a half hour. Set a time that is a bit of a challenge for her. Anything she does with the clay is fine, of course, as long as she does something. It may take her twenty-nine minutes, though, before she gets into the activity. But next time, it may be only twenty-five. And the next time, she might ask you for the clay.
Send her out in the snow and say, "How many things can you do with snow in ten minutes?" Or, once the weather is nicer, send her out with crayons and a pad of drawing paper, and say, "Draw pictures of the smallest living things you can find in the yard. Make your pictures as detailed as you can."
Next time she reads a book, have her read it out loud and make up all the voices. You don't have to be there to listen to her.
Give her some jobs around the house that *have* to be finished. Not icky jobs (this isn't punishment) - just ones that she can feel a sense of accomplishment about when she actually completes them.
Take three quarters of her toys and store them away. Explain that they will come back later, but that she needs to see what she can do herself with only the ones that are out. (Often, seeing too many toys can prevent a child from playing with any.) Let the toys that are out be ones that she can do things with, rather than ones she just watches or listens to. There are plenty of popular toys that essentially turn a child into a spectator, not an active player, so choose wisely.
Can you have her help you with supper? Not just setting the table, but cooking one food - with your supervision, of course. Have her in on every step while you explain what needs to be done and why. (My mama used biscuits with me.) It might make her interested in asking questions about it and doing a little more herself next time.
Have a family game night, and play a game together all the way through. Give yourselves high fives for finishing the thing, whoever wins, and don't have your mind on the laundry or the to-do list while you play - keep your mind on the game. That's an example you set, by the way.
The idea is for your girl to begin concentrating on one thing for a little while (not too long, but enough). A brain that simply skims the surface of things doesn't know how much fun it is to dive deeply.