We'll usually sleep in the morning of, and plan on sleeping in the next day. We go for 8-10 days, so we have some "wiggle room" to sleep in.
The MK late-night EMH is usually the Friday after Christmas. This year, Christmas is ON a Friday
Anyway... We'll normally sleep in a little on a Friday. By sleeping in, I mean "sleeping in" on Disney terms. We get up around 9am, rather than 6:30am or 7am. This is late for us. We'll do another park in the morning and into the afternoon, head back to the hotel to rest, and then do dinner and Magic Kingdom.
Last year, we decided to enjoy vacation as much as we enjoyed the parks. We got some R&R in when we can - and tried not to stretch ourselves too thin. We were with another family, so we tried to show them everything without wearing them out. On the day of the EMH, we slept in and hung out at the resort most of the day. Around 3pm, we went to EPCOT. We had an ADR at Teppan Edo, so we booked FP+ for Spaceship Earth, Soarin' and the character meet-and-greet. We redeemed our FP+s and then went to dinner at Teppan Edo at 7-7:30. We finished up around 9pm and went back to the hotel to change. We were going to go straight to the park, but it was supposed to get a little chilly, so we decided to change. It didn't get chilly - it got warmer

. But we changed into comfortable sweats and the kids boarded the bus to MK, while the parents stayed behind.
We got to MK around 11pm - which seemed to be the perfect time. We grabbed a coffee from Starbucks and watched the Electrical Parade. The park usually clears out a bit after the last parade and/or last fireworks show. Disney actually had a parade at 12am (if I remember correctly) last year... which was really interesting.
Most families head there for one day or with kids, so they pull a marathon day and cap it off with the first showing of the parade and/or fireworks.. so you'll see a drop off after that. Then, you have the people who chose to watch the fireworks and go on the rides during the parade (or vice versa.) They stick around for the last parade/fireworks show. Then they clear out. Some people usually stick around for the EMH, with some leaving by 1am. In our case, the EMH went until 3am. That 1:30-3am (or 2am, in your case) is money. Between 1am and 2am, you'll be able to walk on the Goofy Coaster (we're big kids,) it'll be around a 10-minute wait for Thunder Mountain, Space MAY be around a 30 minute wait (at most.) Those are estimates based on my Christmas trips. The last hour or half hour is fun because the park is almost empty. You have the diehards walking around, but in the area connecting Adventureland to Frontierland, you can go long stretches without seeing anyone. It's very eery near the Haunted Mansion.. Best of all, though, is that you can get up close and personal with the castle and the Mickey & Walt statue. Main Street won't be EMPTY, but empty enough to grab some great pictures. Nothing beats an empty MK in the early morning, but seeing Main Street all lit up (especially during Christmas) and having clear access to the statue offers some AWESOME photos.