Honestly? You both screwed up. You should not have accepted the car and made the initial payment without seeing her sign the title over to you, and same day going to the secretary of state and having it transferred so it would be finalized. You finalized the payment without finalizing the transaction.
On the other hand, the person who sold it to you screwed up by not having the title free and clear.
The only problem you really have is if you want to get some money back out of it.
Personally, I'd follow Dave's advice. Assume you can never sell the car, and assume you're never getting your money back.
When the car becomes utterly broke, tow it back to their house, because it's legally their car and you cannot legally give it to charity, another person, or even a junkyard without the title, and you don't have it.
Let them know that you'd like the title, but you're going to drop the matter. Make sure they understand that since they haven't transferred the title, then they may end up with it being towed back to their place once you decide you can no longer use it. Indicate that this is not what you want to do, but legally you have no choice.
The harder problem will be convincing your wife that this is what's best, and truly having her drop the issue. Unfortunately this is one of those little things that burns people up years later, and ruins years of family gatherings because one and/or the other person are unwilling to forgive a debt.
If it were my wife, I'd simply say, "Look, they screwed us over. It's over and done with, and they aren't going to bend. The reality is that we should not have purchased a car without a signed title. Yes, they took advantage of us, but we let them do it, and at this point they aren't going to budge. We can either be angry at them and be stuck with a car we can't sell, or we can forgive them and be stuck with a car we can't sell. At this point it's our choice, but it's going to color our interaction with them for a long time."
Sorry you're in such a bad position, I know it's not fun. But consider cutting your losses, learning your lesson, and moving on.
---------- Post added at 05:24 PM ---------- Previous post was at 05:16 PM ----------
Oh, one more important thing. If you crash the car, guess who the insurance will pay? You might want to check your policy.
Also, go to carmax.com and get a free vehicle history report, which should tell you if the title is clear or not.