First, I looked at all 262 pitches he has seen that were 90 MPH and over. There are at least a couple sliders mixed in there, but we will keep them in there, ignoring classifications for now.
90-91 MPH, 41 pitches, 12 balls put in play, no swinging strikes
91-92 MPH, 60 pitches, 11 balls put in play, 5 swinging strikes
92-93 MPH, 43 pitches, 15 balls put in play, no swinging strikes
93-94 MPH, 40 pitches, 9 balls put in play, no swinging strikes
94-95 MPH, 31 pitches, 3 balls put in play, 3 swinging strikes
95-96 MPH, 16 pitches, 4 balls put in play, 2 swinging strikes
96-97 MPH, 18 pitches, 4 balls put in play, 2 swinging strikes
97 + MPH, 13 pitches, 3 balls put in play, 2 swinging strikes
He has handled above average fastballs very well, at least when you look at contact versus whiffs (which stabilizes faster, so we don't have to worry too much about small sample size). So it would seem, from this data, that his bat speed is fine. He can catch up with good fastballs and at least make contact with them. Of course, we haven't looked at where he is putting balls in play, which can give us an indication of bat speed.
Here is his spray chart (all pitches) in 2013 so far, according to Texas Leaguers:
Here he is so far in May:
Konerko pulled a lot of homers last year, and he isn't doing the same this year, at least not so far. If you look at his average batted ball distance according to Baseball Heat Maps (teasing out balls hit less than 150 feet), he is hitting the ball over 13 feet shorter per batted ball this season. He is simply not hitting the ball as hard, this is not a BABIP issue. When you look at where he is swinging and missing at pitches: