Please, remove that battery as soon as possible.
If the battery is not charging with the charging adapter connected then it would seem the battery has failed. Keeping that battery connected can damage the circuits in the laptop.
A new battery is likely required, and we always suggest only the replacement batteries recommended by the laptop's maker. Cheaper 3rd party ones have caused problems in the past, with some batteries catching fire.
The next big clue is Vista to XP after a reformat. Although you say you installed all the correct drivers we know from others here that installing XP on newer machines is often problematical. However, if you tell me that you used the Dell web site with this machine's Service Tag to get all the drivers for XP for this Vista machine then perhaps that would be fine.
If you never created your own Admin capable account I fear you have problems, because as soon as you create a new Admin account, (Start > Control Panel > Users, Create new account), this "System Administrator Account" will hide and become unavailable when you reboot and log in to the new account. It will still be available in the system's Safe Mode, but that is no way to use this laptop except for troubleshooting and diagnosing problems.
That means all your programs and settings will need to be reinstalled and re-made respectively. But with XP you will at least be able to access all your personal files for the System Administrator Account in the "Documents and Settings" folders.
Those problems aside, I think you need to do this because with only one Admin capable account your use of this XP laptop is on a knife edge and if anything happens to this account you are stuck with only the Guest account within which you can do very little.
The guidance to do that is in the link below from Microsoft. But I add my own suggestions below because you are still using the System Administrator Account.
How to copy data from a corrupt profile to a new profile - Use the section that starts at, Create a new user profile on the workgroup computer and not a Domain computer (which is part of a network, usually in an office, and administered centrally).
The purpose of this is to treat your current and only Admin account as corrupted because of this Active Desktop problem. If removing the dead battery doesn't solve the problem, (I don't see how it would), then this is the last option I can think of besides a "Repair Install" or reinstall.
1] Backup all your personal files. Creating a new account shouldn't lose those, but how many times has that been said only for something to go wrong!
2] If you use email client software, (Outlook, Outlook Express, Thunderbird, etc), instead of web mail, make a note of email settings, and for good measure export your email folders to some external device. Don't forget your Address Book/Contacts. Microsoft mentions OE anyway in that link, so this is also a preferred step.
3] Backup, (export), your browser bookmarks and any passwords/login details.
4] In User Accounts create Two New Admin Capable accounts. Why 2? We need one to copy the profile to, but we need the other to actually do the work. So for the sake of these steps I will call the first account Admin1 and the second Admin2 but you choose your own names.
5] Reboot and log into Admin1. As soon as you get to the Welcome Screen you will see that the account you've been using is not displayed.
6] Follow the guidance in the Microsoft article from "Copy files to the new user profile" to copy everything from the System Administrator Account to Admin2.
7] DO NOT delete Admin1 as suggested by Microsoft. Having a 2nd Admin capable account is a good idea if ever the main account you will now use ever has problems.
However, I would recommend you password protect all accounts, including the hidden System Administrator Account. Don't forget all these passwords. It may be that no-one other than you and your husband uses this system, but even so, using passworded accounts is still a good security measure. You can add passwords to all 3 Admin accounts in Users in the CP.
I hope all of that helps, and particularly that it solves the Active Desktop problem.