I needed to look this up for study so, given how difficult it was to find this value, I'm mentioning it for future Googlers:
According to "Chemistry of Amines," in section 4, "Important Reagent Bases", the pKa of the N-H proton on the conjugate acid of lithium diisopropylamide (LDA) is 35.7.
wow. i can't believe that information was not readily available. thanks for lending the rest of us a hand.