I've mentioned it to my mom, she says that she won't call the credit card company to fax the affidavit. She didn't state why.
Send the fax it or give up on ever getting your games back. This is absolutely the next step at this point and if you don't do it you're not going to get any more help from them.
What you could do if you still don't get your games
after you send in the fax is to try sending in the same information by certified mail. This requires you to have the information that you would have faxed in the first place. Your last resort is suing in small claims court. Of course, you can't do this either because it will probably require you to use a fax machine to send them the legal complaint, as well as to submit the information you would have had to fax them in the first place as evidence in court.
In other words, I guess you're screwed. Valve has not asked you to do anything unreasonable here. It sounds like the issue is between you and your family, not you and Valve.