The ground pin is tied to neutral at the breaker box, so it is perfectly possible for noise injected into the neutral wire to appear on the ground wire as well.
Beyond that, most recent (not slow) powerline ethernet adapters use 2x2 MIMO, and utilize both line to ground and neutral to ground as the two MIMO pairs. If your adapter has 3 prongs, you can safely assume it is using all 3 conductors to transmit data.
Now, as to what is causing your degraded speed, it is the LEDs. Unfortunately, LEDs when powered from switching converters (like your little 12 V 500 mA brick almost certainly is) are far from ideal loads. Their exponential V-I curve (current draw unusually sensitive to voltage variation) along with their parasitic capacitances generally interact particularly poorly.
What you get is a perfect storm of wideband harmonic noise that gets injected back into the AC mains and the spectrum of this noise is particularly heavy in the 150 kHz to 30 MHz band - the same band that is used by powerline networking.
In fact, it is so bad that according to this research paper, just three 1 W LEDs powered in this manner can cause a 0 dB SNR condition in the frequency band used by powerline networking. In other words, if your signal is merely degrading in speed, you're doing pretty well, considering there is potential for the noise to be so severe so as to cause the connection to drop out entirely.
You have a few options, though without buying a spectrum analyzer to get a peek at the actual spectrum of the noise, it is difficult to know what solution might yield the best results. The main issue is that noise in this band tends to be difficult to filter out for the same reason that powerline networking likewise uses this frequency band - it is hard to block.
That being said, your degraded connection indicates that your SNR is dropping but isn't so low that there isn't hope. You just need to attenuate the noise getting injected some, and you'll see an increase in speed as you attenuate the noise more and more.
I'm going to list some solutions in order of easiness. And they can be used together for extra effect. Doing one might be enough, or you might find that nothing you do can mitigate the impact on your connection speed to your satisfaction. Like I said, it is hard to say without taking some measurements. But with some luck, you might even get away with just the first item on the list:
1. Use an extension cord. Yep. Just plug the LEDs and their power brick into an extension cord. The longer the better. The inductance will help attenuate the high frequency noise some, and the longer the cord, the more the attenuation. It might help or it might do very little but it is so low-effort, might as well try, right?
2. Put it on a different circuit. If you're clever, you can double up with #1 by using a extension cord to plug it in to a different circuit (but keep the LEDs where you want them).
3. Add some ferrites. Search for cable ferrites on Amazon or your online vendor of choice, they are widely available. They clip around the power cord (or better still, extension cord) of the supply powering the LEDs. Put an additional one so there are ferrites on opposite ends of the cord for more attenuation.
4. Add an RFI filter. It can be difficult to know how well these will work, but it will work better when used with cable ferrites than alone. These are also widely available on amazon, though usually you will have to wire directly through them. You can find similar ones that have a proper plug and outlet by searching for power conditioners. Ones that specifically mention 'RFI' as opposed to just 'EMI' filtering will probably work best (RFI implies higher frequency filtering), but a regular EMI filter will help to some degree.
5. Just try a different power supply. This isn't the hardest, but I put it last because it probably won't help. It is worth a shot though.
Unless... you have an old non-switching wallwart laying around somewhere. You know the kind, big, heavy, ugly, from the early 1990s. The important point is heavy. A 12 V one of these might cure your problem completely - inside they have a big old 60 Hz transformer and a bridge rectifier and a filtering capacitor. These won't cause the high frequency harmonics when powering the LEDs, but they also might have poor regulation, and could shorten the life of the LEDs. And the LEDs will flicker. At 60 Hz. Like fluorescent lights. This is item 0 because it is an option but probably not one you want.
Good luck! You might just get away with the extension cord trick. Definitely give that a try if you have one that isn't being used. Your mileage may (will) vary though.