Rescued a baby sparrow - not eating

Hi - we're tending to a naked baby sparrow in hopes that a rehab organization here in southern California will get back to us about taking it in. We found him on the sidewalk, no nest nearby and waited to make sure his mother wasn't coming back for him before taking him in. He's been with us two days now in a small nest we made with a towel and toilet paper and we've been feeding him cat food soaked in hot sugar water (after it's cooled down) every 30 minutes. We have a heating pad on the lowest setting wrapped in a towel underneath the nest and he's in a quiet place. Yesterday he was fairly talkative and hungry, but today he hasn't spoken at all and has only eaten once since 7AM. We've tried feeding him but the food just sat in his mouth so we removed it. Any thoughts on what we can do to increase he chances of survival? He's still moving around in the nest but just seems more lethargic overall. We've left messages with several centers here in LA - one place said he was an invasive house sparrow and they don't take in invasive species, so we're waiting on the others. Thank you!