'Domestic' seems fine to me, meaning that she enjoys housework, doing chores around the house, cooking, cleaning, decorating, etc.
(And yes, 'domesticated' is totally wrong -- that would be an animal that was formerly wild but now lives with humans.)
'Homely' is a huge AE/BE false friend, so beware. In BE, as I understand it, it can mean domestic in much that same sense, though perhaps used more for things than people. (Where in AE we would say 'homey' or 'homelike,' or 'cozy' or something else.)
But in AE 'homely' only means plain-looking, unattractive, having facial features bordering on ugly -- basically a euphemism for ugly. So don't call her that if you're writing to the US. (-:
A homebody isn't necessarily negative, but to me it has a different meaning -- not someone who enjoys housework, but someone who doesn't like going out, who dislikes things like parties or traveling that force them to leave the comfort and security of their own home.
To me a homemaker is the person, usually the wife, who is in charge of the home, who fills the role of primary responsibility for the home and children -- what used to be called a stay-at-home mother. I don't think you could call an au pair that, because it would be usurping the role of a real mother who stayed home with the children. It just would sound out of place.