Why can't you shoot it? If you borrow a .22 and use subsonic rounds it will be quieter than the neighborhood handyman's hammer on a Saturday morning. Also very humane - shoot it in the top of the head in the area between the eyes and ears, on the crown. Sideways through an ear works well too.
We all have our personal hangups on how to deal with this. I have a personal aversion to drowning them. Raccoons are adept swimmers and probably don't die quickly.
My pump up pellet gun is my best friend in those situations. You can get one for under 100 dollars. They will pump up enough to kill just about any animal that will fit in that trap.
Another option is to fill a garbage can with water and drown it. No, it's not pretty but then neither is the sight of the same animal killing your chickens.
I'm glad you were able to repair the point of entry. When you can afford it, electric fencing doesn't have to be anything fancy. You'll need a fencer, a roll of wire, some step-in posts and insulators. I'd run 3 strands of wire at 4", 12", and maybe 24". (We also had a 4th strand at about 40").
Have any power tools? Drill would work but can get messy! Electricity works well too, but can get smelly, well the good old fashion knife slice the throat make sure it is sharp before doing this in the cage. Hey if you have an air compressor and framing gun. No folks I am not a sico just trying to give them an idea, and yes most people don't think this way and if they do they wont say it. Just saying I was a solidier so sometimes you have to come up with CREATIVE ways to eliminate your enemy
To those who are recommending drowning the raccoon - do you realize what a horrible death that is? It is not quick, by any stretch of the imagination, not to mention that it takes a long time for the animal to wear itself out trying to get out of the water until it finally has no energy / air left. Drowning an animal is a horrible idea that causes unnecessary and frightening torture of the animal. In the U.S. it is considered animal cruelty, even when done to a wild animal, not to mention an immoral act of hated toward a living, breathing creature. There are many humane ways to dispose of the animal - several of those have been mentioned here - so please do not drown an animal!!
In many communities the animal control folks will loan you a trap to catch wild animals that come onto your property. Once the animal is caught, you can take the trap - with the animal in it - back to the animal control facility and they will decide what to do with it. They may re-locate it or euthanize it humanely, depending on whether or not the animal is diseased (rabies.) This option would allow the OP to do away the raccoon on their property while letting the experts decide on how to dispatch it. The animal control folks would not need to access anyone's property since you set the trap yourself then return it. It's worth checking into.