"Dear Mister President"

Let me be blunt.
Senator Mitch McConnell has no intention whatsoever of passing mandatory background check legislation, if necessary he will drop napalm on common decency and his own honor to prevent these sensible firearm measures from passing.
Whatever he said to you, however vague or explicit is designed to buy him time until the next tweet or crisis distracts your attention and permits him to quietly drop the matter.  He won't help you, assuming you truly want mandatory background checks on fire sales because Mitch McConnell isn't on your side. No, if you were smart you'd best figure out how far Chuck Schumer is willing to go then see if you can leverage some pressure on McConnell.
That is IF you are serious...and that is likely the key question here, my guess is, you are not...