For the adapter between the motor/ blade, I'd be looking at somehow re-using the connection that comes with your dead mower. So you don't have to reinvent the wheel.
As for batteries - there are all kinds of free, small batteries out there: alarm companies, mobility scooter sales/service places, motorcycle shops, etc.
Just depends how much time you want to spend searching/waiting for a functional, re-usable one vs. just buying a new one. I like the thrill of the hunt, plus it's more satisfying to make functional stuff with as little money and as much re-use as possible.
But I still think you should totally go the ghetto route and find a beefy cooling fan motor from a junked truck, or even better: a starter motor!
You'll be famous when you post the YouTube video of your hacked starter motor mower with a litle 12v car battery on top and jumper cable wiring.