LKJ,
That style of corkscrew in the pic is the nut low. A waiter's friend would be better, but better still is something that is for some reason called an Ah-So:
You slip the long prong between the cork and bottle and work it down far enough to slip the other prong in at the other side. Then you wiggle the handle back and forth lengthwise, working one prong in and then the other, until it's all the way down. Then you twist (gently) and pull. The pressure from the bottle on the bottom of the prongs forces them together and clamps them against the cork, which is extracted.
These things work brilliantly and they shine especially when extracting old or fragile corks. If you drink a fair bit of wine under cork, it's an essential item. That goes for everyone ITT. After trying it a couple times and getting the hang of it, you will never want to use a traditional corkscrew again.
Edit: It will also fix your current problem, obviously. See if you can Amazon Prime one or something.