I originally discovered this by accident and I will help all you veteran players.
I have made a guide to perhaps explain it better here .

IMPORTANT PLEASE READ: This method MUST be repeated if the game updates. If Wizard101 updates on steam; to log in to website-made accounts, just repeat this guide.

1. Download the Website Version of Wizard101. Located at https://www.wizard101.com/game/how-to-download

2. Once that's downloaded and installed. The installation files from the Desktop Version is what you need. It can be found in the C: Drive under the name "ProgramData". You should have hidden files shown. Because ProgramData is a hidden folder.

3. Open that and you should locate a folder named "KingsIsle Entertainment". You can just search "King" in the search-bar on the top right if you can't find it. Inside should be Wizard101 folder. Go into the folder and copy all the files inside. ('Ctrl+A' then 'Ctrl+C' for a shortcut to copy all)

4. Paste those files in the Steam directory. In the Wizard101 Folder in "SteamApps/ Common/ Wizard101.

5. When you paste it, the computer will ask if you want to replace the files. Click 'Yes'.

Enjoy Wizard101. All old accounts should work with the Steam Launcher.

If your account works... Congratulations! Now you're safe to just delete the KingsIsle Entertainment folder I mentioned earlier, because the desktop version of the game is useless now. It doesn't affect your Steam Installation.

Personal Note: If you've got friends on Steam that want to play with you. You need to understand steam account-made wizard101 accounts are not able to play with veteran website-made accounts. So if you're a returning player, take note of that.