POP century has very few king beds and most of them are handicap accessible, especially for the size of resort that it is:
"There are 2,621 rooms with double beds, 115 handicap accessible with double beds, 81 King Bed rooms, and 63 handicap accessible King Bed rooms". (located this on easy wdw). This is a total of 144 king rooms for the entire resort.
Since preferred rooms are located in sections of the 50s, 60s, and 70s buildings, it looks like only buildings 1 3, 5 and 10 have king beds in the preferred section and there are 33 king "preferred" rooms total. (per touring plans).
- Building 1 has 5 king rooms
- Building 3 has 4 king rooms
- Building 5 has 8 king rooms
- Building 10 has 16 king rooms
Each of the buildings that offer king rooms in the preferred section do show to be "handicap accessible" buildings (per the map on the 1st post of this thread). Out of the 63 HA king rooms at POP, it looks like 33 are in the preferred sections, so if a king room is available during your trip, you may have to accept an HA king room.
Out of the 7 times we have requested a king room, we have gotten a king room 5 of those 7 times and twice of the 5 times that we got one, we moved after 1-2 days into our trip, once one became available. Out of the 5 times we had a king room, only once was it an HA room, but we always book standard.
IF a king room is your most important request, personally, I would make sure this is the ONLY request on your ressie. This will allow the room assigner to find you a king room in any preferred location in the resort. Once you start requesting certain area's (50's, 60's 70's) or specific floors, it will limit them to what is available.
If you haven't already added this request to your existing reservation, just call the Disney reservation line and ask them to add it

especially since king room is not an option during on-line check in.
I hope this helps and here is some

pixie dust for the perfect king room