Delicious Broccoli Cheese Soup

I came up with this from ingredients I had in my fridge that I didn't want to waste. All the other ingredients are staples I keep in my pantry or freezer. It was fantastic.

Part of a roasted chicken from the store that no one was interested in any more
Leftover cooked broccoli on the verge of mushy
half an onion
stalk of celery cut into a few pieces
2 tbsp. butter
2-4 tbsp. (?) flour
1 can evaporated milk
Knorr powdered chicken bouillon
black pepper
shredded cheddar cheese
shredded parmesan or 4-cheese blend from Trader Joe's

Put the whole chicken in a pot with some water, the onion and celery and let it simmer until the onion is mushy. Strain out all the solids, leaving just the broth. If desired, you can put the broth in the fridge or freezer to turn the fat solid so you can strain it off, or use a fat-separating tool such as a special cup or "mop". Pick off whatever chicken is left on the bones and put it in a bowl. Put the broccoli in a blender with some of the broth and blend until it's finely chopped. Make a white sauce by melting butter in a skillet or saucepan, gradually adding flour, then adding canned milk. Gradually add broth to make it the consistency you desire. Add chicken bouillon if desired, season with black pepper, stir in as much cheese as you want until melted. Add broccoli mixture and cooked chicken.