Wow, you sound just like me! I have an almost 2.5 year old that sounds just like your older one and an 11 month old who is the calmest, happiest baby on the planet!
I also have trouble entertaining our older one. She is very needy and everything you mentioned about your daughter is just like mine. In fact, she's clinging to me right now!
One thing that I found that she enjoys, and keeps her entertained for more than 2 minutes, is pretend cooking. I pull up a chair at the counter and give her a bowl, spoon, and "ingredients" like flour, salt, marshmallows, anything cheap really. Then she makes me stuff.
But...that lasts 15 minutes at the MOST and it's back to having to entertain her.
Our daughter is very advanced in age and I "quiz" her a lot as a means of something to do. Like I'll count to 4 and ask her what number is next. Same with the alphabet.
Have you bought her any board games yet? She's probably old enough to play a modified game of candy land or something like that.
Since she has trouble staying occupied for long periods of time, just don't stress yourself out about it. Just have LOTS for her to do. Make a list when you have a spare minute (ha ha) of ALL the things you can do. Make playdough (the making the playdough part is one thing to do and the playing with it is another thing to do), read books, watch 5 minutes of TV, color, etc. Come up with LOTS of stuff. Then, when she is bored, you have a handy list right there to reference instead of having to think.
Now that I thought of the idea about the list, maybe I should do it for myself!
Good luck and hang in there!
A little more...now that I thought of it. Not to worry you, but we have had help from a psychologist for our daughter. I'm not sure what realm of not being able to be occupied she is in, but our daughter's psychologist told us just to keep an eye on her due to the possibility of family history, of AD/HD and/or Bipolar Disorder. So I don't want to scare you or have you run to a professional, but if you feel your daughter's attention span and clingy-ness is out of the "normal" range for kids her age, you might want to look into it.