coccidiosis

I believe I found last Sunday on my fire escape a pigeon with
coccidiosis. Initially I thought the feral pigeon in the frozen snow
with the awkward walk, drooped wings and tail and inability to right
itself from its back was suffering from broken limbs or having been
frozen by the recent 30 degree overnight temperature drop. Warming the
bird did not change these findings but demonstrated all the symptoms
Levi lists for coccidiosis, except for the excessive water drinking. I
started giving the bird Bactrim dissolved in water by eye dropper. The
bird has remained alert, awkwardly ambulatory and with a good appetite
for solids. Defecation has reduced from 20+ times a day to three times a
day. But the bird still has all the initial neurological and muscle deficeits.
Is there any possibility this bird will recover so it can continue feral
life? If not, what is the most humane way to dispatch it?
Thank you.
e.rosbooks