When you are doing other things what is her attitude? Is she demanding your attention, waiting for you to get done so you can play with her, sitting doing nothing? I think play dates is a good idea. Are you an at-home mom? If you are, I think having your own priorities for part of the time is a good idea, reading, exercising, visiting with your own friends on the phone and in person. Start with short periods of time. DON'T hand her something to entertain her with while you are doing your thing. The idea is that she can get bored enough that she seeks to entertain herself, right?

I've been here. For me, when my children randomly get into things, that gets my attention because I know that the kids are quite capable of making a mess in two minutes that will take me two hours to clean up, right?

One of my favorites (look back and laugh moment) was when my little boy poured a box of breakfast cereal down the carpeted stairs, followed by what was left of the milk, and handfuls of sugar. There were, by the way, plenty of things available that he could have played with that wouldn't make a mess. Another time, he was (I thought) playing quietly in the other room. When I tried to enter the room, I discovered he had (at age two) quickly pulled every book he could reach off of the bookshelf and stacked them in front of the door, barricading himself in. I had to climb through the window.

Are you sure you want her to entertain herself? I decided that it was worth it, so I rarely lost my temper over these kinds of things. You have to keep a sense of perspective. This is part of the job of a caregiver, Mom or other, to provide the opportunity of independence by calmly dealing with the consequences.

The stairway cereal mess I cleaned up by myself, the books, my boy "helped", he probably put away one book for every five I put away. We sang and laughed while we worked. I was surprised to find that once I decided that being mad was a waste of time, I could genuinely enjoy the cleaning up together time.
Have things that she can mess with, and give her attention when she plays with those things, just a comment on what she is doing, and go on with your activities. Don't direct her to play with them, just give attention when she does. It may interrupt her, but it will help her become more independent.

(added later)
Also, this could just be developmental. She may be "firming up" her connection with you. Children often seem to regress just before they make a developmental leap. If this is so, you need to be more cuddly while she is more clingy. I don't know how long this stage would last. Until she is done with it, I guess.