Here's my thought. Your job doesn't fit neatly into any predeterminted category, but that doesn't mean it isn't a great fit for someone.
If you can take diapers off the general list, however, and are open to being flexible to when the person works, I think you could find a great person by advertising for a housekeeper.
We actually made this transition after our daughter went to pre-k. We were in a nanny mindset, so we created a job with legal pay, benefits, noncancellation guarentee, PTO, etc. We got tons of applicants. It took a while to find someone open to expanding what a housekeeper generally did, but, we found a great person and she's been with us for two years.
She works 15 hours/wk, divided over 3 days. She can set her own schedule and 80% of her time she accomplishes "whatever needs to be done" at her own pace and schedule. This might include housework, laundry, organzing, etc. About 20% of the time, I might ask for specific tasks, or to run an errand, or even to watch my kid while I run out. (I work shiftwork, so I'm often there during the day)
As far as pay, we offered $18/hr (that's what we paid our nanny) Our housekeeper negotiated $20/hr with no PTO (we guarentee not to cancel, but she can take unpaid time off as she wishes)
I used care.com to find her.