Can I modify this LED Light Circuit board to not strobe and just stay lit?
This Q&amp;A has potential value in showing how one MAY go about answering a question like this - or trying to.
IF it works them adding to the answer may be useful as a general guide to show the logic.

NB - All care, no certainty. YMMV: 

Q1: What colour are LEDs? White or red or ?
Q2: What voltage is across each LED when on?
They look like white.
Looks like 9V internal supply - so maybe 3 LEDS in series in 4 strings. (3S4P)
COULD be 2S6P. 

There will be a pin on U1 that goes high / low in time with flashing.
Maybe 2 or 4 pins.
There will (probably) be a pin on U1 that goes high / low in sync with flashing.
It MAY connect to R7 lower which probably connects to Q2 left lower.
IF SO then connecting Q2 middle upper will probably turn lights on.
There may be flash signal on Q4 left lower - if so, connecting Q4 mid upper to ground may turn lights on.
Same for Q3 and Q?1 at lower.
These are (almost certainly) transistors and the single lead on one side is PROBABLY the collector or Drain and PROBABLY drives either LEDs directly. or just maybe via another Qx device.   

So IF flashing signal occurs on these pins on one side of Q1 2 3 4 devices and IF it is low/ground when LEDs are on then grounding them will probably light LEDS.
Hopefully :-)