My 3 1/2 Yo Daughter Will Not Play Independently

I would start by leaving her alone for maybe five to ten minutes a day. She may scream, and be unhappy, but let her know that it is going to happen every day, possibly at around the same time. Leave her in her play room, and let her know she can not come out until you come to get her. There should be consequences if she tries to escape. When you go in to get her, no matter how unhappy she is, smile and be cheerful, ask her if she had a good time, and then leave it until the next day. You can slowly add five minutes about every week, and soon she should learn to entertain herself. This is an important skill that every child should have, so I wish you luck in whatever you choose to do.

Does she have role playing toys? A kitchen, plates, tea set, dolls, strollers, ect.? have her make you something to eat, read her baby a book, wash her baby's laundry, take her baby to the store, ect.
She wants to be just like you. Teach her how. If that still won't work, I have switched to cleaning with vinager. It is non-toxic (edible even) and all natural and CHEAP, but it still disinfects. Then she can clean next to you. give her a dry paper towel and let her go to town. Folding clothes, let her fold a towel. That should keep her busy enough for you to get the rest folded.
Give her some paper and crayons. Maybe a glue stick and some pieces of string. If you set things at her level, a whole new world will open up to her. She will learn.
Good luck.

I don't have a 3 yr old, but I do have a one year old...so I can only give a little advice/outlook.
Right now, I'm guessing, you're the coolest person she knows. Of course she wants to be by your side all day because she wants to be like "mommy". She wants to be a big girl. My advice is to let her help out like she wants to. If you're doing dishes, let her "dry" the plastic bowls or cups (nothing breakable so that she can't hurt herself). If you're doing laundry let her help fold and ask her if she wants to help mommy.
Paying bills...let her push a couple of the keys on the keyboard if electronic...if writing, get her to help you "hold" the paper still so you can write.
All she wants is to help mommy and be like you.
My mom said that when I was about 2 yrs old, I was helping out with practically everything she did during the day...everything from dishes to helping her "wash" the floor...mind you, i was using a dirty rag to do that, but hey! I was helping.
Embrace this period, as I will in a couple years...she wont want to be around you soon, let alone help with simple things around the house.
Good luck and enjoy!

Find a good game or something she likes (coloring, paiting, playdough, blocks) and sit down w/her. Let her get into it, then quietly slip away. This usually works w/my 3 yr old. Or you can give her things to do around the house while you're cleaning. My daughter loves to clean windows or sweep while I'm cleaning too. I usually just fill a spray bottle w/water which yes, I have to clean it up later, but she goes off on her own and I can get in some good cleaning at the same time!

S. ,
One thing you can do is give her little jobs to help you have her get all the trash in the house bring it to you ,then have her but containers back,if you recycle have her help you there,Then there is the V_Tech video system for 3yrs and up it's games with all there favorite cartoon friends ,and this will help them learn coler shapes,letters ,GOING outside everyday is also very good for them let her plant some flowers and tell her she has to take care of them every day .
well i hope this helps
L.