Should I keep this pigeon or let it free?

If it can fly reasonably well, set it free.  

If it still can't fly, you can call a wild bird rehabilitator to care for it.  If you don't know any, call the veterinarian, they should know.  

I have taken in a pigeon that couldn't fly.  It was able to fly within a month, so I took it back to where it came from and released it.  Normally a pigeon can fly miles to its home, but I didn't want to make it fly that far when it had just recovered.

If it had taken longer to recover, I would have released it where I was, because it would probably think this was its new home.

If you think you should care for it, it should be in a cage or something it can see out of, not shut up in a box.  You would need to get pigeon or dove food, or wild bird seed, and give it water.  It might do better on the bottom than on a perch, because pigeons like flat places, and anyway you don't want it to fall with its hurt wing.  

I suggest you take it to a rescue organization, you can contact your local animal shelter for a referrral.

I was at the beach with my dogs at sunset one night and there was a pigeon acting strangely near the waters edge, it kept flying out over the water then back to the beach.  There were dogs all over and I thought it was going to get grabbed and killed.  When I approached it it didn't seem afraid and when I got close it looked like something was going on with it's eyes (it was getting dark so I couldn't see well).  I was able to walk right up to it and wrap it in my jacket.  

It was a bit of a trick getting it to my office with my three dogs in the cab of my truck trying to check it out, but when I got it into the light I could see someone had put a single stitch into each of it's eyelids and literally sewn it's eyes shut.  Then I remembered this creepy fat guy who had "jogged" past me four times while I was carrying the bird, he was probably the one who had done this.  

I left it in my office that night in a quiet corner and the next morning took it to a wildlife rescue.  I was afraid to hurt it by removing the stitch.  They pulled out the stitchs, kept and and gave it antibiotics for a few days and released it.

The best thing you can do for an injured bird is get it to someone who knows what to do to help it.

Call the library and ask them to give you the contact info for a local wildlife rehabilitation center.  Make sure that its the type that'll take-in birds.  

You can do both.

If it can't fly and is badly hurt, keep it for a month, feed it fresh water & wild bird food and see how it is in a month.  If you can't care for it, give it to a rescue.

Take it to an animal welfare center or call the DNR

Unless you have an aviary or something where the pigeon can rest, let it go. It's probably just stunned hitting a window.

You should set it free.

It's a wild animal so keeping it in captivity, especially when you're not really able to help it, might be doing more harm than good.

Let it go, they don't last long without food or water