Windows XP was succeeded by Windows Vista in 2007, which was in turn succeeded by Windows 7, Windows 8 and Windows 10, so the chances of getting any modern software to run on an XP machine have to be pretty slim.
The latest version of Adobe Digital Editions is v4.5.8, which was only released in April of this year, so its highly unlikely that the developers intended it to be compatible with either Windows XP or Windows Vista. You almost certainly need to install an older version of the software. A question similar to yours on the Adobe forum suggests that v1.7.2 (which will have been issued not long after Adobe Digital Editions was first introduced in 2007) will run on an XP machine. You can download the installation file here:
http://kb2.adobe.com/cps/403/kb403051/attachments/setup.exe
If you'd like to try a more recent version, the download file for v2.0.1, which was issued in 2013, is here:
http://download.adobe.com/pub/adobe/digitaleditions/ADE_2.0_Installer.exe
and the equivalent file for v3.0 (issued in 2014) is here:
http://download.adobe.com/pub/adobe/digitaleditions/ADE_3.0_Installer.exe