To make it easier for my daughter to put on her mittens, and also to prevent losing them, I bought those mitten clips so you can attach mittens to coat sleeves.  They broke within a week.  Then I thought of a simple idea: I tied some yarn to both mittens, and made it long enough that it will go from one sleeve to the other, staying inside the coat, so the mittens will stay put.  The mittens have plastic loops so you can attach them together, and I tied the yarn to the loops.