Daycare is pretty much the same as far as activities and age groups go, though daycares often take infants from 6 weeks and children to 6 years thanks to short mat leave and late school entry.
Daycare places are rare as hens' teeth in some cities, and you have to pay. There is no such thing as free nursery (or free daycare) unless you fall into a very low income bracket.
I doubt many daycare places would be happy to accept a child for only two mornings per week or so because they might then have to turn away someone who wanted a place for the full week, or chance someone else arriving who wanted all the other hours your DD didn't want. They are businesses and they have rent to pay for their premises and staff to recruit and pay.
Daycare is very much for working families needing care for their children. Families where one parent is at home usually provide enrichment activities at a variety of places that run classes for toddlers and preschoolers - libraries, YMCA, some museums, music schools, etc. Families that want one on one care for their children while parents work hire a nanny or au pair. It's kind of not the done thing to send your child to a daycare if you are at home. Being able to spend time with your DCs and go to classes with them or just plain veg out with them when they are young is considered somewhere between a luxury and a privilege.
If you want a couple of hours to yourself, advertise for a babysitter in one of the many universities or even in a good high school - DD1 made a lot of money babysitting when she was in university in NYC. Many young American women spent their teen years babysitting and they are good at it - if you do this, you can ask for references, and be prepared to spend time phoning women in North Dakota or Chicago or Louisville - and many students like to connect with a family to babysit and might have a few daytime hours free, depending on their class schedule. Students who are far from home may especially like to see an actual home as opposed to a dorm, and a family, and little children can make a welcome break from roommates.