Hey, niceguy. The pigeon's definitely in pain, even some minor wing breaks won't make them sleep on their side like that.
About pain medication:
You have two easy options: Asprin, or Ibuprofen.
(DO NOT USE TYLENOL/ACETAMINOPHEN - It'll kill the bird.)
It's only going to take a tiny bit to help the bird. and I mean TINY. We're going for about 1-2 mg per dose.
(the drug formulary says 5-10 mg per kg, and a bird is usually about 1/3 of a kg.)
That's about 1/100th of a single pill. Yes, that's ridiculously small.
So, what we can do here is take the easy route and pick up a small bottle of liquid "children's advil" or "children's motrin". If you can't find these, ask the pharmacist for "an ibuprofen suspension, nothing else in it." This should be pretty cheap, especially if there's a generic. Again, stay away from anything with acetaminophen in it! No children's tylenol!!
(you might want to print out / write this down, and ask the pharmacist, because I've probably confused you by now.)
Then we can help you do the math to dilute the liquid medication with water enough that the pigeon doesn't get a full child's dose of the medication. A child is a lot bigger than a pigeon.
You'll want to get 2 or 3 drops of this into the bird a couple times a day. This is pretty easy - if you take a stick and drip a tiny bit along his beak, he'll instinctively swallow it to get liquid off his beak.
For those interested in the math:
I've done some early estimates, and it looks like the standard children's liquid advil / motrin preparations contains 100mg / 5ml = 20mg/ml . both brand and generics seem to use this standard.
So that's 20mg of medication per 1 ml.... and it turns out both the pharmacist's drop and the kitchen drop come out to 1/20 ml. Handy!
http://en.wikipedia.org/wiki/Drop_(unit)
So we can estimate there's about 1mg of ibuprofen per drop of medication. That's ridiculously convenient.
Please double-check my math?
There's also some pretty awesome wrap bandages that do a good job of sticking to themselves, no pins or tape needed - pharmacy should have those too. You won't need more than one. Once there's some pain medication in the bird, we can go ahead and try to bandage up the wing in that closed position, before the bone's healed in the broken position, okay?
And yeah, if you can get a helper to help turn the bird over, it'll make wrapping the wing a lot easier for you.
Do what you can to help the bird, we'll do what we can to help you.