If the bleeding's stopped and the bird is looking reasonably alert (not drooping or huddled up) that's a good sign. Don't worry if it isn't eating -- you can supply water, but an adult bird won't be too much inconvenienced if it goes overnight without eating. Shelter from extreme temperatures, and quiet surroundings, are more important for now.
It's been awhile since I helped out at the veterinary hospital's wildlife clinic in my old town, but I'll try to remember the time our avian specialist talked me through an "arrival exam".
--pick up the bird using an old towel. (Gloves are handy, especially if you have a disposable pair, but a towel can be wrapped around the bird so it doesn't flap and injure itself further.) Covering a bird's eyes can be a big help in calming it down.
--we would do a thorough initial exam at the start. It actually stresses the bird less than a bunch of shorter repeat handlings.
--handling a bird that might be ill is actually safer than handling a sick mammal -- yes, there are some risks, but not as severe as hantavirus or especially rabies. (All the people handling mammals had to have preventative shots -- the avian team didn't.) Some of my co-workers did get grossed out when fleas or other parasites crawled onto them, but unless the bird has a huge parasite load, you might not see anything at all. If you're really worried, use disposable gloves and a breathing mask (or a kerchief tied over your mouth and nose) -- but I worked at that place for 2 years, saw everything from pigeons and gulls to eagles, and nothing serious happened to me or my colleagues. (Well, I got my hand scratched by a ferruginous hawk, but that was about it.)
--start at the bird's head and gently work your way down. Its eyes should be bright and clear. Look for obvious open wounds, feel for broken bones, and note any unusual amounts of dirt, discoloration, etc. If you're seeing a lot of parasites, the bird may have been unwell for some time. Discharge around the beak and eyes could also be an indication of illness. Look inside the bird's mouth, to check for blood, or signs of infection or fungus. Wooziness and blood around the beak might result if it hit a window hard and sustained brain damage. (I found a beautiful rose-breasted grosbeak with this once ... usually this is fatal.) Stains around the vent could mean digestive upset or pesticide poisoning.
--one thing I've found helpful is to feel the bird's breastbone. A sharp, unpadded breastbone can indicate that it hasn't been getting enough food. (Sometimes birds can get so weak in the winter that they just can't fly. This would happen quite a bit to juvenile birds of prey around this time of year.) Also feel the crop -- sharp objects or hard masses could indicate it's eaten a foreign object or the food's become impacted in there.
--extremities -- look for damage to the feet (frostbite, infection, etc.) -- and feel the wingbones. Fractures are usually pretty obvious. If they're in the bones above the "elbow", this will probably need professional attention (the muscles pull fractured bones out of alignment). Fractions in the radius/ulna area below the elbow can sometimes be splinted by amateurs at home. The vet told me that minor damage to the "fingerbones" (below the bird's "wrist") may heal up on their own, without splinting -- and that a bird might be able to fly away after a couple of days of recuperation. Bird bones take a shorter time to heal up than ours do (they are lighter and have a faster metabolism).
Good luck to you and "Walter"! Droopy birds can sometimes revive if they are kept warm (e.g. with a heatlamp) and quiet, overnight. The good thing about birds is -- they tend not to linger, and generally pass away quietly within 48 hours if the situation is beyond remedy.
p.s. this page is for baby pigeons, but does give some tips for feeding sick adults.
http://members.aol.com/duiven/medical/feedbaby.htm