Power demand from the board (w/o other peripherals like HDD, etc) is not likely to be much more than 20W. The CPU is only 8W.
True, but the Intel D945GCLF2 board uses the 945GC chipset wich is a power hog at 22.2W TDP.
So from 8W CPU +22 W chipset to 35W CPU and 6W chipset (H61/H77), 11W more should be no problem for the power supply.
Can you remove that black heatsink block?
I think the Kozuti should work, unless you stress the CPU 80%-100% for long stretches of time.
I found two 1155 ITX board with 5 or more Sata ports: Asus P8H77-I
and Zotac H77-ITX WiFi
. Zotac H67 board has 6 Sata, too, but I don't know if it supports i3-3XXX CPUs. I would buy Asus.
With both boards you have the option to add a seperate drive for your OS, mSata (SSD only) for Zotac, Sata for Asus, something I would consider. My HP Microserver runs Ubuntu Server from a USB stick. Without the fan swap (no time yet) I envy the silence of a passively cooled system.