Monkeybird - it's really hard to advise you on specific systems and storage without knowing your house / life. However, you do have to first decide where everything has its home - you cannot run a tidy household if things don't have a place to go.
To take your examples, this is what I do:
sewing kit and sewing machine - sewing kit is in an Ikea storage box in the hall cupboard (which is huge, with shelves) along with another 20 same stacked boxes containing all craft things ie there is one box for ribbons, one box for crayons, one box for paints, one for needles and thread, one for buttons... I don't have a sewing machine (there is a clothes mending / alterations man 20m from my front door and I send all mending/alterations out)
breastpump and bottles - have never had these, always found breastfeeding so much less clutterful
kids toys - there are many! - in children's bedrooms, in boxes in their toy cupboards or on display. Some in the dining/play room, but they get put away and/or exchanged for others pretty frequently
incoming letters - on hall table (me) or on hall desk (my partner)
kids artwork etc - in a cupboard specifically allocated to this in the hall
spare batteries and lightbulbs so I know where to find them - in a huge Ikea storage box in my housemaid's cupboard (there is another box for spare plugs/adaptors/extension flexes)
pushchair - in hall cupboard
many many many books CDs DVDs - I throw away/donate/recycle lots of books and CDs and the rest are stored in two huge bookcases, one in the dining room and one in the hall. Children's books are in bookcases in their rooms
Monkeybird - do you think that your furniture meets your storage needs?