Lindsey Graham has described choosing to be Trump's VP as liken to "Buying a ticket on the Titanic."
In short, nobody with a sound mind or an interest in their electoral future is going to agree to it. Gingrich has neither, so he'd be my best bet.
I actually think Gingrich could be a great choice for Trump. He would be able to coach Trump on policy, and also legitimize him to many conservative and establishment skeptics. I mean, yeah, he has skeletons in his closet, but I dont think they'd come up too often.
Gingrich's only real weakness is that he's an old, white man.
The 2012 race showed that any respect or legitimacy that Gingrich has from the establishment evaporated sometime in the last decade. He's now a Sunday Morning talking head who is able to sound smart for short periods of time before talking about moon bases or having poor children do janitorial work in schools.
He does nothing for a Trump ticket, certainly not electorally, and if he's chosen it'll be because he was the biggest name willing to say yes.