EDIT: It can be done, see my second edit below.
Fundamentally can't be done. This issue was parodied at XKCD. The gist of the problem is this: you can't duplicate the firepower of the sun, especially if you're using a solar-based power system that isn't 100% efficient. Even if it was. It would need to acquire 100% of the solar energy that would hit the moon during a full moon, transfer that power perfectly (100% efficiency... the engineer within is starting to weep) to LEDs, which can emit the collected power as photons with 100% perfect conversion (oh, the pain!).
Can't be done without serious power. Serious. Check out the link. Serious.
Also, remember that where there's solar panel, there isn't LEDs. You can hide the batteries underground and put the panels on the backside of the moon (it's tidally locked), but that means you must capture and store enough power to illuminate all those buka-watt LEDs for each night. Serious.
OK, Shadoweze has piqued my curiosity. Lunar albedo for a full moon is 0.12. Albedo is the ratio of energy received to energy reflected. The sun bathes the moon in 1kw/m. So the reflection, what we need to achieve, is 0.12kw/m.
The full-moon lunar surface is 1013 m2. That means we need to generate 1.2E12 watts or 1.2 terrawatts. The most efficient solar panel in 2018 has a 22.2% efficiency. That means for every kw of solar energy we'll actually have only .222 kw to work with. That's twice-ish what we need, so far so good.
Average lunar light is about 0.015 foot-candles or about 0.0019 lumens per m2 for 0.016 lumens-per-watt of lunar emittance. 2018's most efficient LED is 105 lumens-per-watt. Good news! We don't need to cover every inch of the earth-side of the lunar surface!
Better news is that for each pass of the moon in front of the sun, there isn't a commensurate "full-moon" pass for the earth. The moon varies from a new moon (100% LED use) to a full moon (0% LED use). I'm absolutely wrong with the assertion I'm about to make, but to keep this from becoming a full dissertation, let's assume we only need to store 50% of the power needed to hold a full moon all the time and there's enough space between the LEDs that we can use detectors to shut off the LEDs we don't need during each phase of the moon. (And I'm ignoring the fact that we only need to turn the LEDs on when the emporer is in the night cycle. Who cares what the peons see, right?)
OK, I'm convinced. Shadowzee's right. It can be done. It might need enough battery mass to shift the moon's orbit... but it can be done.
Why is the XKCD no longer relevant? It's emitted light from the earth reflecting off the moon. That takes a ton more power, and I'd ignored it.