The power cable normally winds up on a spool. Well, it managed to wind up OUTSIDE the spool inside the body of the vacuum and get tangled up. Sort of like a bicycle chain coming off. I see this happen to retractable badges/measuring tapes quite often.
First I tried to open up the vacuum. I removed all the screws I could see (4 on the bottom and 2 inside the dust bag bay), but it was still held together around the wheels. I even took a wheel off but couldn't find any more fasteners. There must be clips inside.
Fortunately, I found out the plastic socket for the power plug can be pulled out:
The problem wasn't over yet. As you can see, access was limited. I used a chopstick (because it wouldn't damage the cable, unlike a screwdriver) to push the cable back as best as I could (the spool is pretty big), then turned the spool by hand to play out the cable. Fortunately, this was enough to correct the cable (again with the bicycle chain, it's like hooking the chain on the start of the gear and turning the pedal so the whole chain wraps around the gear).
The final situation: