After the BIOS was working again, I closed the case and started monitoring temperature and fan speed...

CPU: 60° C.
Case: 44° C.
Fan speeds almost supersonic. ;-)

Not good.
So, how do we fix it?

First of all I ripped out the case fan that came with the case.
It made noise, but didn't really move air.

I had some fans lying around, so I used two of those. (There is room in the case to install 2 fans.)

Instead of relying on the CPU fan, I used some cardboard to funnel the air over the CPU heatsink when air is extracted from the case. I also disconnected the CPU fan as this isn't needed anymore.

Yes, I know it doesn't look very nice, but it completely works:

CPU: 43° C.
Case: 32° C.

Much better, but still noisy...