Hey there!
I had the same processor running at stock with an 280mm all in one and it still hit 80c+ especially when I don't run my window AC. Before you drop 70 dollars or more on a cooler, try undervolting and looking up some guides. My 9900k was hitting 1.35volts or higher with stock motherboard settings and even the VCCIO and VCCSA was at 1.3volts for some reason. I settled on 1.22volts for the cpu at 4.7ghz and dropped the other two voltage settings to 1.10volts. My 9900k loads around 70-75 celcius when my apartment is at its hottest. Just food for thought before you spend all that money.