B737 Captain reported deviating and exceeding 250 kt. below 10;000 ft. after departing LAX following a wake turbulence encounter.

Narrative

I'm authoring this report to address an inadvertent airspeed deviation on LADYJ4 Departure from LAX Runway 24. The initial takeoff and rotation were normal. ATC issued us a climb to 8K; negating the initial departure at-or-below altitudes. During the climb-out and flap retraction we encountered mild wake turbulence from the preceding B737; necessitating a slight push and roll to counter those effects - not severe enough for the upset procedure. This occurred a couple of times during the flaps 5 to flaps up sequence; to include immediately after clean-up. Throughout those corrections in the climb; I was able to maintain a centered Flight Director (FD). At a location approximately between ENNEY and EYENO; while flying the FD command bars; I noticed the airspeed at 280 KIAS. A quick confirmation confirmed there were no airspeed indicator issues. I immediately increased pitch to bring the airspeed back to 250 KIAS. As I did; the FD initially indicated a slow speed indication; fly-down; followed by an adjustment to on-speed; both FD bars centered; as I reached 250 KIAS. The remainder of the departure was uneventful and no concerns were directed to us by ATC.At level-off my FO and I tried [to] de-construct the event in the debrief. We both remain unsure what caused the momentary deviation. Perhaps; it was the slight push; unloading; to prevent an upset during the climb-out - which seems to make the most sense. Perhaps; it was changing winds during climb-out - winds on the surface were variable. Neither of these explain the centered FD command bars. We experienced no other issues or concerns with the FD on this flight or the next.