My ds (now 2) was like this for about 5-6 months from the age of 13 months, and like you, it drove me up the wall. We called her 'limpet child'.
However, she has got so much better over the last 4-5 months, that she might be a different child. I do remember wondering if it was ever going to get better though.
Some of the things I tried were:
Spending a few mintues showing her how to play with whatever toy she was interested in, then moving away to sit down. I didn't make a big thing of moving away (for example saying, "Right now mummy's going to have a cup of tea" - as I used to do, since that always seemed to remind her that I wasn't playing with her anymore. I just quietly withdrew (although at first I had to stay within sight). I realised that in the spirit of letting her discover her own imagination, I had been guilty of sometimes not showing her how to play. We tend to assume that babies will discover things for themselves.
I found this quite hard to do as the temptation was to rush off and do a few chores, or pop to the loo, but gradually I found she became more and more happy to sit and play for a while by herself. It might have been a question of only 10-15 minutes, but it seemed a breakthrough for me.
The other thing I did was make sure that I spent regular 20-30 minute periods throughout the day focussed entirely on her. I am quite an extrovert person, and love going out for coffee mornings, but I suddenly realised that although we were together for the whole day, I was perhaps not spending that much time just with her. My daughter seemed happier to let me do my own thing every so often, if we'd spent some time reading together or building towers etc.
My last point though is that you shouldn't beat yourself up about this. I think it's something that is common with most eldest children (have I interpreted your circumstances right?). It's at about this age that they start to become aware that you are a separate person from them, and the constant need to check you're there is part of the (albeit uncomfortable) separation process.
PS I'm tremendously impressed that your ds does so well being at nursery 3 days a week. I put off sorting out childcare until my ds was 2 as I was so worried about the clingyness. Interestingly she took to it no problems this year. I suspect she might not have done so if we'd tried 6 months ago.
Good luck