Since you labeled the question as "science-based", I have to warn you that no scheme you might implement will stand up to any scientific analysis:
Your cave can't exist
Simple as it is. The location you've chosen is 11km under the ground, so the rock is under extreme pressure. Under these pressures, even rock becomes plastic, so any cavern will collapse over time.
Also, even if your cavern does not collapse, it's still in a region where rock is being bent (because the pacific plate is forced under the Asian plate (rock that's plastic!)). So cracks will show pretty much wherever the rock is too hard to flow. And you've got a water pressure of more than 1.1 metric tons per square centimeter (1100 bar). This pressure will press the water into any sizable cavern.
Your light lacks an adequate energy source
The power of the sun is astounding. It provides us with
1.3 kW/m^2 when it's in zenit. Ok, let's say you settle for some softer illumination, say
100 W/m^2. A square kilometer has a million square meters, so for an area of
2.5km x 4km, you already need
1 GW of power. That's a nuclear power plant. Just for lighting an area of
2.5km x 4km to levels that don't even come close to what our sun provides.
Your cave lacks adequate cooling
Even if you managed to handwave the power source, you still have the problem of the power drain. You see, those Gigawatts need to go somewhere. And if you don't provide for a way for them to reach the outside, the energy just stays in form of heat. Did I mention that the rock is already typically quite hot at these depths? Great, now you are adding even more heat.
You cannot believably use sea water for the cooling, because, as pointed out above, any water near your cave spells doom to air breathing life within.
You may write an interesting story with your deep underground cavern. Yet this story will be more in the fantasy realm than in the (soft) science-fiction realm. Hard science-fiction cannot contain your brightly lit cavern.