Looking for someone who can search all midrashim/chazal/yalkut etc for original source If I had otzar or bar ilan, I'd do it myself, using search words like I typed below from kitov's hebrew text. I'm assuming rabbi kitov didn't make it up, but culled it from some authentic midrashic source.
Here's loshon from Sefer Hatodaah:
ומה היא גבורתו שהיה יכול לעמוד ביסורים יותר מכל אדם שעל פני האדמה לכך נתנו הקב"ה באמצע הדורות שיהא נושא עליו כל הדורות שלפניו ושלאחריו עד שיגלה כל כחו לעת קץ הימים כקורה זו שנותנים אותה באמצעה של התקרה כדי שתעמיד את כל התקרה
Although it's also possible that it was indeed his own conclusion,, and if counting the "midpoint beam" from Dovid's birth, the Geulah would have been in the mems' (1984?), a year which would have sounded plenty futuristic and eschatological, from rabbi ki tov's perspective (like g orwell's lhavdil)
On page 142 of "The Book of our Heritage, Vol 3" by R. Eliyahu Kitov
[King] David however was the greatest hero of all the generations. What was his strength, that he was able to withstand suffering more than any person on Earth? G-d placed him at the midpoint of all the generations, so that he should be able to bear upon himself all the preceding generations and all those following till his power would be revealed at the End-of-Days -- like the center beam which holds aloft the entire ceiling.
Using this information, and the date of King David's birth of 2854 (from Chabad.org ) as well as the fact that he lived 70 years we take the midpoint of his life and double it to get the year of the Geulah:
( 2854+35) * 2 = 5778