Does it have to be the 9600?
Yep. The 9600k is sitting on my desk right now. Motherboard and ram in the mail.
I'd guess you're probably not buying new but the new prices of the 9600 and 10600KF are very similar on PCPartPicker.
Got it for free from my nephew.
If you can't make it work for the B580 then a 2080 for $200 would offer excellent price to performance, though you may find yourself limited by the 8 GB VRAM before too long. If you can find a 6700 XT for no more than $260 or a 2080 Ti for no more than $275 that would offer about the same price to performance as the B580 (based on TechPowerUp's database of relative GPU performance). Knock about $30-40 off those prices to have price to performance on par with a $200 2080.
I'm stuck with the 9600k as it was free. I'm paying $90 total for the mobo, and ram. That's practically half a computer for $90 bucks. Adding a CPU into the mix would make this upgrade a non starter. Plus it's 30% improvment just in CPU upgrade. Id love to get a reasonably priced 2080!