My daughter is 22 months old and stopped napping a week ago. I feel your pain.
To do my emails: I bought her a second-hand V-Tech kids' laptop on eBay. Stops her interfering with my computer.
To cook: I sit her in her high-chair, a safe distance from the cooker, and give her playdough and cookie cutters, or split peas/lentils and plastic cups to play with.
To get the laundry done: I ask her to help me load the washing machine with clothes. Takes longer, but it keeps her quiet and she's happy when she thinks she's helping.
To make a phone call: we have two phones, and I disconnect one and let her talk on that while I talk on the other one.
To do paperwork: I sit her next to me in a highchair at the table and give her pencils and paper to scribble with.
To do the washing-up: I put her on a step-stool in front of me, keep the water temperature warm, not too hot, and let her splash about in the bubbles.
She is very domestic, and when I have to sweep the kitchen floor, I give her the little brush from the dustpan and let her rearrange the dirt while I make sure the worst of it is dealt with. She's also happy with a damp cloth: she thinks she's cleaning the house. Running water has her fascinated, too.
Oddly, I find it a little easier now she doesn't nap. Now I don't have to make sure we're at home at naptime: there were so many times she fell asleep in my arms on the Tube and I had to carry her home. She used to nap right in the middle of the day and it made arranging to do anything quite awkward. I'm trying to look on the bright side.
I've always been a stay-at-home mother but I've decided to send her to nursery part time, for just two afternoons a week, to give me a chance to do the many DIY jobs that are piling up around here. I'm not going to let her help me tile the bathroom.