I did my quilting in a 5' X 8' walk-in closet for several years. It had double folding doors. I bought a used 3' X 5' wooden table for my machine and put it on one side. I bought a used wooden computer desk that just fit my 24" X 36" cutting mat. The pull-out for the keyboard held my portable ironing area (Walmart). Bought those plastic shelves to put beside that for my fabric. I had plastic bins under the computer desk and it had a drawer with a cabinet on one side. I stored batting and extra machine under the sewing table. I could sit in my rolling chair and reach everything!
I would split that room into two sections with a gate inbetween. Use a desk for your sewing machine with drawers. Put a 3 to 4 inch board across the back of the desk to keep things from falling off the back and place it so you can see your children and they can see you. Use shelves along the walls and a fold-down ironing board...perhaps on the back of the door? Put a toybox on the children's side and teach them to put up their toys before leaving the room. Take frequent breaks to read or play with them. You can even purchase small roll-out mats for nap time when they get fussy. Be sure the door is on your side of the room, so they can't leave while your back is turned or you are distracted! Hope you can work everything out with your room. As the kids get older, you can expand into the rest of the room!