We stayed at the DLH from January 5th through the 8th. We book a standard view room with our AP discount which was $149.00 a night and when we got there to check in I asked if any of the rooms had been complete and were ready to stay in. I told the lady I was really hoping to stay in one because I had seen the pictures online of the renovated rooms and they were stunning and not only that but we were celebrating our 5th Anniversay and had our daughters with us one of whom we were also celebrating her weet 16 which is a big deal.
The lady told me to hold on a minute and walked to the back and when she came back out she informed me that she had gotten permission to upgrade us to one of the new rooms with a pool view on the 8th floor of the Dreams tower

I was super excited and the room was beautiful!!! Disney really did these rooms very nice and put in some great detail! They were able to upgrade us for free because the hotel was only at 30% capacity, which was great seeing as how there were some post on DIS boards and other disney sites saying that anyone going in the 1st week of January was in for the largest crowds of the year......I can tell you it was the very opposite of what was predicted and it goes to show that the information given was only an opinion and information like that is not always correct.
I had a feeling anyway that the information about crowds would be wrong seeing as how we have gone to DL for the past 6 Januarys and have always had light crowds in from about the 4th of January through the 15th.
Anyhow just ask about the rooms and I am sure if capacity is low they will try to make it happen for you! Best of luck and have a great trip!