I have an Asus EB1007 EeeBox that doesn't have a CD drive. I removed the hard drive and connected it to my laptop and formatted it, via a USB to SATA/IDE adapter.
How can I install Windows XP using this adapter?
You need to install Windows on the EeeBox.
The easiest and fastest method, given Windows XP, would be to use WinToFlash to get the WinXP setup on a USB disk/key. Then you should be able to boot the EeeBox from the USB disk/key and install Windows from there.
Alternatively, you could get an external CD/DVD drive and use your adapter with it, but it's much faster and less expensive to go with a USB key.
Put the hard disk back in the Eee Box, then use the USB-to-IDE/SATA adapter to connect a CD drive to it. You may need to poke at the BIOS options to get it to boot from what will be treated as a USB CD/DVD drive, but you should be good to go from there.