I rsised a female baby house finch from about that age. We got her baby parrot formula and kept it warm and slightly thick. She would take it from a dropper down into the area in her throat that had the bright marks inside to help the parents hit the right spot. I think we fed her every hour or so and over night if she started screaming for food. After a while she would sleep through the night and get fed during the day when she demanded it. She slowly self weaned and bonded to us so we decided she needed to stay with us.
She was happy, stole our food and would bathe in any low filled water bowl while we held it.
She played with her toys, sang, and got a year and a half she wouldn't have.