I had 4 kids, and it was easier to just train them to keep up with things rather than doing major weekly clean ups. When they were toddlers, I'd pick up with them, but once I knew they'd mastered the skill I made sure they did it and did it right each time they were asked - whether I was right there or not.
I limited toys and things, too. We weeded out toys regularly and I kept about half their things in a large closet. I left the other half on shelves & in toyboxes for them to play with. Every couple of weeks we'd go through the closet and take out things they wanted to use, but for every toy, game or video that came out of the closet - one went in. It worked quite well for us, and by the time the kids were school aged, they did their own rotating and could choose to keep more in their rooms if they wanted, as long as they took care of them.
Once you're sure that cleaning her room is a realistic chore for your daughter, and you're sure she knows exactly what's expected of her, it really becomes a heart issue. When she plays instead of cleaning her room she's choosing to disobey you. She obviously knows it's wrong because she gets to work as soon as she hears you coming.
If she was my little girl, I would go into her room with her, give her a couple of specific tasks (pick up the legos, put the Barbies and all the Barbie clothes away, etc.) and tell her that I wanted those things finished when I came back in 5 minutes. If the tasks were finished when I came back, I'd praise her, tell her how much better things are looking already and give her a few more tasks to do in the next 5 minutes. When her room was finally tidy, I'd make a point of complimenting her on her great work and telling her how pleased I was that she hadn't wasted the whole afternoon because now she'd have lots of time to play or do other things. I'd probably offer her a snack or treat, too, but I'm an easy touch.
If she hadn't the tasks I'd given her, I would take those toys and put them away for a few weeks. Then I'd give her her next few tasks and another chance to obey quickly. If she's stubborn, she might end up with only a few toys in her room, but she'd know that I meant business. After cleaning her room that way for a few weeks, she'd have the skill and desire to pick up her things without so much supervision.