There are so many options that this question is a bit unclear.
#1 - Don't vent. You have a window. Already meets code.
#2 - Run a bulkhead go out where you want.
#3 - Buy a kit for your vent to make the top of your vent unit run straight up (bigger). Use that space to figure it out.
Your question isn't about how to vent. It is how to vent and it looks nice. There are tons of custom tops for vents that hide this stuff or you can drywall that part. I see almost a 5" gap that you could go straight out. Sorry this just has so many options not really sure what you are looking for.
Also usually 6" is required for a round duct by most inspectors. But given my point #1 you really have no requirement so you can install a 5" duct. Or you can get ducts in "code" sizes like the 3.5x10 - just an example I have used I am sure there are a multitude of sizes... There are lots of options. You do not need an 8" duct unless you are in a commercial kitchen.
Update: Given the last round of pictures your question is much different. It was, what can I do... now it is holy crap this won't work because I have a finished kitchen. So given how nice your kitchen is finished and the fact that you would be INSANE to disassemble this at this point there is only one solution.
I have a local metalworks shop build you a custom vent adapter that will fit this and go out the 5" opening. They will just need to fit it to the 8" pipe and honestly it shouldn't be a big deal to do. (labeled #1 below)
The only other option is that they trim out the top - create a larger metal box/soffit and go out label #2 below. This could look great or be an eyesore. This will probably be costly to do if it looks nice.
Don't vent, you have windows right there.
Buy a different venting system that allows a smaller output or one that has a larger top so you can go out a bigger area. The vent should have been hooked up and ran before finishing the wall. Period. You cannot be picky about getting exactly what you want now unless you are willing to really open up that wall - might mean losing all of the tile - or at least a big big chunk of it.