This can be extremely complicated or pretty simple. The basic process is that microbes, earthworms, or other things eat carbons (like wood shavings) using nitrogens (like chicken poop) as fuel. Some people go to great lengths to get a perfect balance of carbon and nitrogen, but a lot of us don’t. A lot of people call this process composting but I just think of it as rotting.
In the future one very simple way to handle this is to do like Boskelli said, empty your coop in the fall after the garden is finished. By spring it will have rotted and be great nutrition for your garden. Another way that you can make really complex or extremely simple is to compost it. You can try to balance the ingredients, keep the moisture level perfect, and turn it regularly. This will give you useable compost in a few months. My method is a lot closer to piling it up and letting it rot. It takes longer but unless you have heavy equipment to turn it, turning that stuff is WORK! I use the two pile method, one working pile and one storing stuff to become the working pile.
There are two potential problems dumping that stuff directly in the garden with growing plants. If you get too much nitrogen (the poop) it can burn certain plants if it comes into pretty close contact with them. Certain plants are more susceptible to his than others, tomatoes, squash, cucumbers, and such. Some things like corn can take more but you can still burn them. The other potential issue (and not likely to be a real problem, still I’ll mention it) is that too much carbons can suck a lot of the nitrogen out of the soil so the bugs can use it to break down the carbon. The nitrogen is not lost, once the carbon is broken down the nitrogen is back in the soil, but it is “borrowed” for a while and not available for the plants.
So where do you go from here? I think you’ve pretty much done it. The problems happen when it is too concentrated, either nitrogen or carbon. You’ve mixed the soil so the concentration isn’t very high. The bugs that eat it need moisture and you’ve been watering it. You don’t really want it sopping wet, but just keep it damp. Damp is the best condition for those bugs to work. I don’t know how much longer you will wait to plant something in there, but by reducing the concentration and letting it work a bit, you should be fine. I’d just pretend I knew what I was doing all along and carry on. Just don’t put any fresh manure on it but start your compost pile.
Good luck!