I've had to do this with critically injured mice and a budgie who was suffering and would not survive a vet trip, its CO2 euthanasia and doesn't sting or feel cold like the dry ice version:
Supplies:
1 box baking soda
1 jug white vinegar
plastic container with tall sides (like a medium -- large critter keeper, or a big mayo jar)
towel
Glass liquid measuring cup, or coffee cup
Large plastic bag that wont leak
small hose (I rob the one off my fishtank vaccuum)
Take plastic container and add something soft inside, this is where the chick will go so make her comfortable, put in the bird.
Place folded towel over top. Place one end of the hose in the container.
Take plastic bag, pour half a jug vinegar inside, then fill measuring cup 3/4 full with baking soda. Set measure cup gently in the bag but dont tip it into the vinegar.
put other end of those hose in the bag and wrap with tape or just hold it tight.
grab the side of the bag and tip a bit of the bakingsoda into the vinegar, the bag will fill with CO2 fast, squeeze it gently into the tube so it goes into the plastic chamber, 30% is anesthetic
Chick will pass out, then pour more bakingsoda in the vinegar bag and flood the chamber with a higher concentration of gas to euthanize. Leave towel on top for a while, the gas is heavier than oxygen and will stay in.
Its kinder than freezing, and you dont have to chop their head off. It seems easiest on the birds because how efficient their lungs are, the little budgie shut his eyes immediately. I want to add this is nt for animals over 2lb ever.