[Nyororin]

Even as a tourist, you should be able to rent a place through somewhere like Leopalace. You may disappointed at the size of the place though. They do offer 2 bedrooms, but they are rather scarce. Your best bet is to look into 1 bedroom places and all sleep in one room. You`ll need to pay for the rent and like in advance in one lump sum.

At 1.5, kindergarten is out. Kindergarten in Japan is 3+, public and private.
Public hoikuen (a combined nursery school and kindergarten) accepts children 3 months and over, but unless you are in a very lax area (with tons of spaces for kids open) you will need to be a resident and both parents working. As a tourist, both of these are impossible, even if you are able to rent a place.
You shouldn`t have any trouble with private daycares. The cost can be really prohibitive with those though - especially if you are short term and paying by day. It generally runs around 30000yen a month for around 8 hours a day. Otherwise, around 500yen an hour.

As for location, pretty much anywhere in Japan would fit that. With the long-term living plan and needing to rent a place to stay, I would say to stick to fair sized cities and their outskirts.
I personally recommend Nagoya area, as it`s a bit city with a lot of resources yet low priced, plenty of day cares and karate dojos, and a decent number of family type monthly rental rooms.
But - as I said - most places in Japan outside of Tokyo and Osaka will work.

ETA; And while I was away eating and taking too much time to type this message up more information appeared...

You probably won`t be able to renew inside Japan unless you are on a working holiday visa that is split into segments.
I have never once heard of a daycare center being called デイケア in Japan. There is "デイ・ケア" but it is always used to refer to care of those with disabilities, and never for child care. (Unless the child has a disability that requires specialized care.)
You`ll want to look for 保育所 / 保育室 or 認可外保育施設 to be more specific.